I would not designate it as a part time job on your resume. On the application itself, you'll probably have to specify how many hours you work at that company, and this, or the overlapping dates, will signal to the adcom that you are working two jobs at once.
I'm sure there are tons of applicants facing a scenario like this. I'm actually in the same boat with my startup. Coming out of undergrad I was working full-time at my startup, but since then I jumped at an opportunity to take a management position at another company (in addition to running my startup). My hours are very flexible at both companies, but generally I'm working 20-30hrs per week at my startup and 40-50hrs per week at my other job. My startup is listed 2nd on my resume since it's chronologically dated prior to my other job.
The dates certainly will overlap, but I don't see this as an issue at all. It's not like you are lying or being misleading. If they can't read between the lines, it's nothing that a short one sentence explanation in your interview can't solve.