Am I parsing this incorrectly, or is it really possible to read in ambiguity without more context?
Yes, I think this is ambiguous indeed, and 1 is a legitimate reading.
Another less ambiguous alternative (you mention one using "by"):
Uber is a flexible way to earn extra money and meet new people [,] while giving rides with your own car.
I looked at it and I don't feel "own" really removes the ambiguity.