The guy I'm with now introduced me to his dad after about 2 weeks and his mom the day after I met his dad. I'd met his parents, sister, aunt and 2 cousins before he met my parents. We've been "together" for about 3 months now and he still hasn't met my mom, but that's because she lives on the other side of the country.
When I met his dad, we were out at the bar and he was there too. I went to dinner at his mom's when I met her and out to dinner when I met the rest of his family. As for him meeting my mom, it just so happens he will probably be meeting her at her 50th birthday party, since she is coming to visit for it.
He and I aren't even technically dating yet, due to some strange circumstances, but...we're still sorta serious if that makes sense. We moved fast in our relationship so I don't know. It really depends on the situation and when it feels right.