How to understand "don't make us look like jerks here" in this context
Solution 1:
I think the first option has the correct sentiment, but I'm not sure it's necessarily about trust, as such.
I would phrase it more like this:
I believe you are responsible enough to have a bike - don't make me look like an idiot by proving me wrong
Solution 2:
I would say the first option. They don't want the son to do something unfortunate and make them look like fools for trusting him.