Jenifer - I'd say yes! We absolutely recommend doing reference checks. I don't recommend contacting HR though (I say that quietly as an HR practitioner). When we do reference checks, we ask many behavior based questions on key issues. For example, if you need to know how they'll handle conflict situations, ask "tell me about a time when this individual had to deal with a conflict and how did it go?" And, pay attention not just to the answer but the non-verbals.
Whether it is full-time or part-time or exempt or non-exempt, we'd highly recommend you do them for both.
If you'd like to talk further, just reach out to me (Robin@strategicHRinc.com)
We still do them, too! We have dodged a couple of bullets that caught resume fibs or inflated duties. After an interview that goes well, we ask a candidate for a list of 3-5 professional references and reach out to them via email or phone. Most of the time we get "I can't say anything" but one response was colorful enough that I let the candidate know they should probably vet their references before sending them to potential employers. (She was not hired)