I am trying to visualize$\ z = f(x(t), y(t))$, and my model is that$\ x, y, z$ depend on$\ t$. The only way that I can visualize this is as a point moving in 3D space. However, wouldn't that be $\ \vec p = \vec f(t)$ ?
Is there any way to visualize the relationship between$\ z$ and$\ t$ as a surface that is changing in time, or is this wrong?
No, you should draw one surface $z=f(x,y)$ and think of parametrized curves on that surface. That is, given the parametrized curve $(x(t),y(t))$ in the $xy$-plane, you look up to the surface and watch the particle moving along the surface.
You could indeed end up in $4$-space by including the $t$-axis, i.e., plotting $(t,x(t),y(t))$ and then moving into one more dimension with the plot $(t,x(t),y(t),f(x(t),y(t)))$ — still a curve on the surface $z=f(x,y)$.