Why is it bad that someone can use something like a park or an emergency room without paying?
According to the standard criterion used to evaluate welfare (Pareto efficiency), the problem with 'free riding' is that goods are not produced even though they cost less to produce than they are worth to consumers. That means that we are forfeiting an opportunity to help people without harming anyone else.
To illustrate using a standard example, suppose that 10 people live on a street, and that each would value an additional street light at £100 (i.e. each would pay up to £100 to bring a new street light into existence). The street light would cost £101 to produce, but if produced can be used by all for free: this is known as 'non-excludability'. Will it be produced? If people are self-interested, then the answer is no. Although any resident would benefit from the street light, building it would cost them £101, which exceeds the benefit (£100). That's a problem since the total benefit (10 x £100 = £1000) greatly exceeds the cost of the light. If the inhabitants could be compelled to pay for the street light, sharing the cost equally, then they would all be better off.
I should stress that, from the perspective of Pareto efficiency, the problem with 'free riding' has nothing to do with 'equity'. To use your example of parks, the problem is not that people get to use the park without paying. Rather, the problem is that, since people are expected to free ride, the park won't be created in the first place.
I would also add to @afreelunch's answer that not all free riders play the exact same role in every situation.
In the example cited by him about street lights, it is quite obvious that people there could get an extra profit thanks to those lights. But if they do not buy them, nobody will lose anything (except the opportunity of having that extra profit).
However, let's imagine this example: 2 cities, A and B, which are dumping pollution in a shared river. This pollution is causing diseases in the population, so both cities are "losing", in public terms. The problem could be solved with a single machine next to the river. This would solve the problem for both cities, even in the case that just one of them paid for it. In this case, those 2 cities do not represent the same kind of potential free riders as in the street lights example, since in this case both cities need to do something. In this "coward game", A or B WILL pay for the machine, even knowing that the other will behave as a free rider, because the alternative (not building the machine) is worse not only for the free rider (which would use the public good for free), but also for the city which will build the machine, since they both have a big pollution problem.
There are many other contexts in which "free riding" does not represent exactly the same situation, and as a consequence the potential solution for the game will be different.
The existing answers are covering mostly only situations where the 'free riders' aren't using up anything. It's important to note that an emergency room is absolutely not such an example. Someone who is not paying for an emergency room visit is using up someone else's resources without compensating them. They are taking up an ER room that could be used by other patients, taking up time of the doctors, nurses, and other staff, and using up consumable goods such as medications, gloves, needles, etc. All of this costs money.
As a result of the free rider, everyone else is paying more, receiving worse service, or both compared to what they would have if the 'free rider' either paid their share or weren't present.
There are other aspects, which (I think) has not been mentioned by other answers.
If people have the choice to pay for something, or to be a free rider, almost everyone will choose to be a free rider. If everyone pays their fair share (for some definition of "fair share") of the cost of road building & maintenance, and the roads get built, then everyone benefits.
If Dave is the a free rider, the roads will still get built. In a country with a population of millions of people, everyone else will only need to pay a fraction of a cent more. They wouldn't even notice, right? And while technically, the roads would wear out faster, the amount is also negligible. So you could argue, why not let this one person be a free rider?
There are 2 problems with this:
This argument applies (individually) to everyone. You could argue it for you, I could argue it for me, and so on. The more people that are exempt, the bigger the cost to the remainder. Eventually it DOES mean that people will need to pay more.
If individuals are exempt, then those that do pay, consider this unfair. Why should I pay, if you don't have to? Collecting the money to make roads becomes harder, and at some point, there might be civil unrest by the people paying.
Both of these are issues, for example, with tax collection. Some companies are exempt - for example, if a company has enough negotiating power, they can talk themselves out of taxes (or some part of taxes); if the (local or national) government does not agree, they will go elsewhere. Once one company succeeds, other companies will use the same tactic.
Then, when individuals realise that many companies are not paying taxes, they get (understandably) upset. This strongly links to how fair people think a particular exemption is - if someone has no income, they will not pay income tax, this would not cause people to be upset. However, if someone has millions of dollars in an overseas tax haven, and they pay no tax, or if a company doesn't pay tax because they buy their product from an overseas parent company at the exact price that they don't make a profit, people get upset.
There is also a related cost to potential "free riders" - if the ambulance is paid for by the insurance companies, they might not come to the aid of the uninsured "free riders". The cost of this is also born by the insured - if you are insured, but can't prove it, will you be left behind?
Surprisingly no one has mentioned the tragedy of the commons. Having been born in the Soviet Union where everything was public and everyone was a free rider, I used to call it the tragedy of my life. But anecdotes aside, the problem with free riders is if they are allowed, eventually everyone becomes a free rider. Eventually, such a place deteriorates into, for a lack of a better word, a dumphole. Now, parks and other public infrastructure exist because government enforces tax collection equally and those parks are paid for and maintained mainly using taxes so in reality there are no free riders. There are those who pay, but don't use it. Same with public hospitals that the healthy never get to use, but they do pay taxes.
Public things can exist and be maintained in theory and practice in tight communities where free riding is nearly impossible to hide. Usually in at least seasonally isolated communities, by geography and/or climate. Examples are very small villages around the world or even sparsely populated countries with harsh climate like Scandinavia or Canada. There, a free rider wouldn't be much of a problem for anyone but himself because his identity would likely be a common knowledge and he would have little options to avoid punishment. Hence, an instance of free riding would never get a chance to snowball into the tragedy of the commons.
It is a "problem" because if no one pays for the good or service (they are 'free' riding after all), sellers of that product have no incentive to produce it. If no one produces it, and the good or service is beneficial (e.g. national defense), then surely that is a "problem"? I don't think anyone wants to live in a country with no national defense.
The same logic applies to parks and emergency rooms. If no one pays, there won't be any parks or emergency rooms. If you are thinking that the govt. should pay for it instead, understand that the govt. has to get the resources to pay for it from somewhere. That leads to taxation. There is no such thing as a free lunch. Someone has to pay for it at the end of the day.