Using Parks-Mcclellan (or others) for FIR differentiator?
There is no fundamental difference between the design of standard frequency selective filters (such as low pass, high pass, etc.) and differentiators. The only difference is that for a (linear phase) differentiator, the desired frequency response must be formulated as a purely imaginary function of frequency, unlike standard frequency selective filters, for which the desired response is purely real-valued. But that's a minor issue, and all important algorithmic details remain the same. Generally, algorithms only become different as soon as the desired response must be defined by a complex-valued function (with non-zero real and imaginary parts).

E.g., the Matlab function firpm, which implements the Parks McClellan method, can design FIR differentiators if you supply differentiator as filter type. The same is true for the least squares filter design function firls.

In sum, designing linear phase FIR differentiators is no more difficult than designing any other FIR filter.

Concerning low pass differentiators, take a look at this answer.