Insurance is cheap, bike cost $1,500, oil is cheap (6,000 factory change interval, and truck oil which works well in my bike costs $13/gallon)
Your parent is right, tires can be expensive. It depends on the bike you ride and the tires you buy. I can get ~10,000 miles out of $80 of tire, and I install them myself.
I'd say it really comes down to the bike you have. Some bikes are way more expensive to own than a commuter car, but that doesn't mean all bikes are.