So Rinna and Meishi definitely made that promise to each other. As for his parents being detectives it could be a retcon on the part of the author- that was a minor detail shared in the first ten chapters- or it could be they retired or changed jobs for some reason.
Also I see a lot of people mentioning this but: Rinna being underaged isn't the reason Meishi holds himself back so much. Hell, pretty sure the AoC in Japan is still like 16 or something? But Japanese culture is huge on respect, propriety, and doing things "properly". Which is what Meishi wants to do.
He wants to respect Rinna by being 'worthy' of her and standing as her equal in a way he can feel proud of. He doesn't want to skip steps or rush into things because of that. Hell, remember when they got outed and most of the precinct mobbed him and even wrote "sex-crazed detective" on his forehead? Yeah. He wants to prove (if only to himself) that he's worthy of her, respects her, and cares about her for more than her body.
That's why he holds back despite the fact that Rinna is all but intentionally riling him up in the hopes of taking things further. Well, that and the fact that she is a minor- age of consent or otherwise- in tandem with the implications that when push comes to shove Rinna isn't quite as ready to "find out" as she believes she is.