Is it possible to calculate a z-transform for a filter calculated with Parks-McClellan?
If your impulse response is $h(n)$ then the $\mathcal{Z}$-transform of this impulse response (i.e. the transfer function of the filter) is

$$H(z)=\sum_{n=0}^{N-1}h(n)z^{-n}$$

where $N$ is the filter length, i.e. the number of coefficients.

Note that since it is an FIR filter, all its poles are at the origin of the $z$-plane. The zeros are of course the zeros of the polynomial with coefficients $h(n)$.