DM - I would tell your DP's parents over the phone first. My DP's parents were not at all supportive. They did not come to our wedding and are only marginally accepting of me in the family. DP told her mom over the phone, which I think gave her time to have her reaction (which was very negative), express her opinion to DP (that we should never have done that), and process it before we actually had to spend time together in person. It was much better for me, as the pregnant person, to not have to take in the stress of being around someone who was not at all supportive. By the next time we actually saw them in person, she was able to keep her opinion to herself and interact civilly with me.
As for the work thing, sounds like we do similar work. I am also out to all my co-workers and they're fine with me, but I don't come out to clients. I wear a wedding band, so I assume that they all assume I have a husband. Hardly anyone has ever asked about it. (Actually, since I work for a Catholic agency, many of my clients think I'm a nun! but that's another story.) When they do it's more like "What does our husband do?" And I'm pretty good at talking about DP without using pronouns. Surprisingly, many of them are just now noticing that I am pregnant, even though I've been showing for quite awhile. I don't bring up personal stuff with my clients, so unless they ask, I haven't talked about my pregnancy. Only one person has asked anything about "my husband" in relation to the pregnancy. She asked if the baby was a boy would it be named after my husband. I responded that the baby would be named after his father (which is the truth, we're naming the baby after DP's brother, our donor). I guess for me, I've just decided that it's fine with me if my clients assume I have a husband. And if someday I am stuck and have to use male pronouns to speak about my DP, it's not the end of the world. I've decided that it's not something forced upon me. I could tell them that my partner is a woman, but I am making a professional choice to keep that part of my personal life private.
Good luck with both situations!