Pareto Optimality:
Since preferences are convex in your case, you can find Pareto optimality in the same way. You need to solve $\text{MRS}^{A}_{v,w} = \text{MRS}^{B}_{v,w} = \text{MRS}^{C}_{v,w}$ where $\{v,w : v \neq w\} \subset \{x,y,z\}$.
The reason this works is because you can maximize $U_A$ subject to $U_B = \overline{u_B}$ and $U_C = \overline{u_C}$ and similarly for the other two agents. Since your utility functions are convex, when you set up the Lagrangian, you will find that the MRSs have to be equal to one another.
Walrasian equilibrium:
You have to solve the following (three) maximization problems: $$\max [U_P(x_P,y_P,z_P)] \text{ subject to } p_xx_P + p_y y_P + p_z z_P = p_x w_x^A + p_y w_y^A + p_z w_z^A$$
for each $P \in \{A, B, C\}$. You will be able to find $p_x : p_y : p_z$ and the allocations.
Additionally, you can verify that these points lie in the Pareto optimal set due to the first welfare theorem.
Later edit: Your set of equations for Pareto optimality will be:
- $\begin{aligned}\text{MRS}^A_{x,y} = \text{MRS}^B_{x,y} = \text{MRS}^C_{x,y} &\iff \frac{\frac{\partial U_A}{\partial x}}{\frac{\partial U_A}{\partial y}} = \frac{\frac{\partial U_B}{\partial x}}{\frac{\partial U_B}{\partial y}} = \frac{\frac{\partial U_C}{\partial x}}{\frac{\partial U_C}{\partial y}} \\ &\iff \frac{y_A z_A^2}{x_A z_A^2} = \frac{2x_By_Bz_B}{x_B^2z_B} = \frac{y_C^2z_C}{x_Cy_Cz_C}\end{aligned}$
- $\begin{aligned}\text{MRS}^A_{y,z} = \text{MRS}^B_{y,z} = \text{MRS}^C_{y,z} \iff \frac{\frac{\partial U_A}{\partial y}}{\frac{\partial U_A}{\partial z}} = \frac{\frac{\partial U_B}{\partial y}}{\frac{\partial U_B}{\partial z}} = \frac{\frac{\partial U_C}{\partial y}}{\frac{\partial U_C}{\partial z}}\end{aligned}$
- $\begin{aligned}\text{MRS}^A_{z,x} = \text{MRS}^B_{z,x} = \text{MRS}^C_{z,x} \iff \frac{\frac{\partial U_A}{\partial z}}{\frac{\partial U_A}{\partial x}} = \frac{\frac{\partial U_B}{\partial z}}{\frac{\partial U_B}{\partial x}} = \frac{\frac{\partial U_C}{\partial z}}{\frac{\partial U_C}{\partial x}}\end{aligned}$