The wavefunction covers 3-dimensional space. Formally, it maps each point in space to a complex number. That is, it is a function that takes a point in 3-dimensional space as input and returns a complex number as output: $\Psi:\mathbb{R}^{3}\rightarrow \mathbb{C}$.
There are many ways to represent 3-dimensional space: cartesian coordinates $(x,y,z)$, spherical coordinates $(r,\theta,\phi)$, cylindrical coordinates $(z, r, \theta)$, to name the most common. We have these different ways because functions might look "nice" in one coordinate system and not nice in another.
As it turns out, the wave function for the $p_{z}$ orbital looks "nice" in spherical coordinates, that is, $\Psi(r, \theta, \phi)$. It looks less nice in cartesian coordinates, but you can do the replacement for $\theta$ and $\phi$ as noted in the question. However, people do not usually mix coordinates systems together because they are redundant. For example, no one writes $r\sqrt{x^{2}+y^{2}}$. They write either $r^{2}$ or $x^{2}+y^{2}$.
To conclude, the $p_{z}$ orbital you have written is a function of 3 variables: $r, \theta, \phi$. It just happens not to depend on $\phi$, so it looks like just a function of $r$ and $\theta$. Forget this concept of dimensionality because it's not doing you any favors.