The most useful thing to know is that rapid tests have a pretty high specificity, meaning when they say positive, you almost certainly have the virus - false positives are rare. Your friend's right: asymptomatic infection is totally real and happens more often than people realize. You can absolutely have COVID and feel completely fine, especially if you're vaccinated or if it's a newer variant. The virus is there and you're shedding it, so you're contagious.
Yes, you should isolate, at least for a few days. I get that it sucks when you feel normal, but this is exactly the situation where you could unknowingly pass it to your parents or whoever else you'd visit. People don't always show symptoms immediately either - sometimes they develop over the next day or two, sometimes they never do at all. If you really want to be sure it's not a fluke, you could take another rapid test in a few hours or grab a PCR test if you can, but generally one clear positive line is reliable enough to act on.
The isolation period these days is more flexible than it used to be. Most people aim for at least a couple days symptom-free, though some folks go longer to be extra safe around vulnerable people. Reschedule the visit if you can, or at minimum let your parents know before you show up. Better to be cautious here.