My parents didn't visit, I lived with them! LOL My husband was away at military training out of the state for the last 3 months of pregnancy. He walked in the delivery room literally 5 minutes before our daughter was born, got to stay 10 days, and then had to go back to the training for another month or 2. We all stayed at my parents house after she was born. Some nights the 3 of us were sleeping in a twin bed together! haha
I was also in the middle of my last semester of college with a couple classes. So, I stayed a while at my parents' house so that they could help me out and I wouldn't be alone.
After that, we've probably visited them (or them coming by to see us) almost every other weekend. They are only about 45 minutes away and we are all used to the drive - my dad works in the town I live in & so makes the drive every weekday.
My in-laws is a different story. I never met my FIL, he lived several states away, and he passed away a couple years ago (I never wanted to meet him anyways, he was abusive...). My MIL and one SIL live about 11 hours away. They didn't get to meet our daughter or even ME until our daughter was 10 months old! And we've only been down there to visit them maybe 5 times. I have another SIL who lives way up north & I have never met her. Financially & time-wise, none of us can travel to see everybody very much. We are trying to get my MIL to move up here because her health is declining.
Now, when I was growing up, we saw my mom's dad & step-mom maybe once a year. Like one day a year. And they live 10-15 minutes from my parents' house. They are just odd... lol
EDIT: Just saw your added question... most of the time my mom offers to help out or will invite us to their house. Or I can just ask & they usually don't mind if we come over for the weekend. It's easier for us to go there since they have dogs & we have no where for them to sleep at our place :)
For my MIL & SIL, we have always made the plans to go down to visit them, except for 1 time we invited my MIL to come up to my family reunion (5 days in a lodge on the lake). She doesn't have the means to just come visit us (money, car - she took a greyhound to come up here that one time).