Teaching at a community college while in grad school is certainly an option and I know people who have done it.
My major concern would be workload. I'd expect a community college job to be drastically more work, greater responsibility and stress, and less likely to accommodate your coursework and research time. It can be very difficult in grad school to balance your time and energy among your responsibilities, and it's easy for the short-term demands of teaching to push the longer-term demands of research to the back burner, ultimately sabotaging your chances of finishing your degree. I certainly wouldn't suggest a full-time community college appointment; even part-time seems like it would be hard to manage. But it sounds like you've actually taught at this college before, and have also TA'ed, so you'll be in the best position to make that judgment.
At the university, you'll probably get to TA a variety of courses, including more advanced courses in your discipline. This can be a great learning experience for you, and fill in gaps in your education that you didn't know you had. It can also be an asset on the job market. At the community college, I presume you'd teach introductory or remedial courses exclusively; that's a learning experience too, but not in the same way.
At the university, you're more likely to find fellow students and faculty mentors with whom you can discuss aspects of teaching, and being a TA can be a great social / bonding / networking experience. If you stay in academia, you'll be able to get letters from well-known professors attesting to your teaching ability. At the community college, you'll probably be more on your own, and a letter from the department head there may not carry so much weight.
Overall, it might be better to plan on teaching community college during occasional summers (when TA'ing may not be available), or perhaps near the end of your PhD (and especially if your departmental funding runs out). It seems overly ambitious to plan to do it throughout.