Our situation was a little bit on the extreme side... we weren't planning on telling family until after 12 weeks, but then my Paternal Grandfather got sick and wasn't expected to live very much longer. I wanted to tell him before he died, so we told him when we visited him in the hospital. There wasn't really a way to tell him privately, so my Dad's side of the family ended up hearing the news very, very early. I think I was maybe 6 or 7 weeks along at that point?
Sadly, my Grandpa died a few weeks later.
Our news came out at the funeral (Grandma wasn't thinking so clearly and she forgot that it was supposed to be a secret), so we ended up scrambling to call immediate family so that they could hear it from us first. At that point, we'd seen the heartbeat on the ultrasound, so I was OK with it becoming general family knowledge. Even my Mom, who has some serious jealousy issues with my Dad (they're divorced), was completely understanding. But, like I said, we had some very sad extenuating circumstances.
I could see not wanting to tell one set of parents because you didn't want to burden them with that news. If you think that they might spend weeks 6-12 feeling edgy and nervous, it might be a kindness to keep them from worrying. I think that's a situation where you really have to know your audience. And I don't think that keeping it to yourselves for a bit necessarily precludes you from telling them if something does go wrong. As far as they know, maybe you were just waiting for the right way/moment to tell them.
I do think that, if you tell one set of parents first, it has to be a set of parents that you know can be trusted with a secret. My Mom seems to take perverse pleasure in giving away people's secrets, so once she knew we pretty much had to tell my in-laws. I wouldn't have put it past her to call them herself. Even if I'd told her not to. My Dad, on the other hand, is generally very good with secrets. I was OK with him knowing before anyone else, because I didn't have to worry about him blabbing to the whole family.
I do think that, if you tell one set of parents first, and then the other set finds out that you waited to tell them, you could have some serious relationship
fallout on your hands.