Part time industrial work during a mathematics PhD
I think it would be very challenging and unusual to work part-time at a job and be taking graduate classes or doing thesis research.

But it's quite common for graduate students to work full-time (as interns) at government laboratories or in industry during the summer (or at another time of year) for perhaps 3 months.  Mathematics students I have known have worked at


Google
Microsoft
US Dept. of Energy labs (Sandia, Los Alamos, Livermore, Oak Ridge, Argonne, etc.)
Oil companies (Schlumberger, Total)
US NSA contractors (cryptography)
US Geological Survey
NOAA


and many other places that I've forgotten.  Talk to your advisor and to other faculty in the program.  Your university may also have someone who coordinates these things.