To answer this, the first thing I had to recall, through the haze of years, was *which *of my girlfriends had ever met my parents.
Before my wife, AFAICR, only three of my GFs met my folks. My college GF came down to visit me over spring break after we’d been together a matter of weeks. (It was true lust.) When I was in my early 30s and living in downstate Virginia, a GF and I wanted to do some stuff up in DC, so we stayed with my parents while we were there.
In both of those cases, the point wasn’t to introduce her to my parents, but it was kind of unavoidable if she and I were going to be under their roof. I wouldn’t have brought them home specifically to introduce them to the folks.
One relationship when I was in my late 20s seemed to be getting serious, and she took me to her folks’ for Thanksgiving, and she came to mine for Christmas. This was maybe about 9-10 months into our relationship. (We broke up just a few months later, but meeting the folks didn’t precipitate it.)
And my wife and I knew we were serious pretty much right off the bat. We met in grad school, and she took me home to visit her folks over spring break after we’d been together ~5 months, and I took her up to meet my folks after the semester ended a couple months later. (In this case, the answer to “what’s the rush?” was “I’ve just met this great person that I am really wild about, and I think I want to spend the rest of my life with. I sure hope you like him/her too.”)