I teach physics at a community college and have been on a number of hiring committees for full-time positions, as well as participating in a bunch of hires of part-timers. Different departments and different individuals have different priorities. Some full-timers who have a master's might actually feel threatened by a PhD applying for a part-time position. Often we get applications, for both full- and part-time positions, from people with PhD's that clearly show that the applicant has no real interest in teaching and thinks it should be easy to get a community college job as a fallback. A PhD will typically be somewhat of an advantage in applying for a part-time job, especially if the course to be taught is at a high academic level (e.g., differential equations). The bigger problem is your lack of teaching experience.
There is a lot of randomness involved, and you can maximize your chances if you're in a big urban area and apply for a lot of jobs. Sometimes a full-timer gets sick or there is some other last-minute reason to hire someone to teach a class. In these emergency situations, you basically have to be available, meet the minimum qualifications, and give a non-disastrous interview.
The fact that you want to teach in the evening is a good thing. Many full-timers don't want to teach evening classes, so often those are the hardest to find a teacher for.
I think MHH is right that this discussion is going to be more valuable if we widen it to include full-time positions. For full-time positions, the value of a PhD seems to vary greatly. At my school, for example, the math department has 3 PhD's and 27 people with masters', while the natural science division has 21 with PhD's and 11 with masters'. One department clearly considers a PhD more important than the other does. This may be partly because the math department's offerings are bottom-heavy with remedial classes. For a full-time position, there will typically be a list of minimum qualifications, which are set by law, and a list of desirable qualifications. The two biggies on the list of desirable qualifications are teaching experience and a PhD. It helps if you have both.
When we hire for a part-time position, we're hiring someone to teach a specific course. If the course is low level, then we don't care as much about whether the candidate has a PhD. When we hire a full-timer, theoretically we want someone who can teach every course their department offers, but realistically we usually have something more specific in mind. When someone with a master's is hired full-time, usually that person ends up getting slotted into teaching gen ed courses, remedial courses, or other low-level courses for the rest of their career. Many people are very happy in such a slot, e.g., I hear that many folks in math see teaching remedial math as their ideal job, and they have no interest at all in teaching calculus.