Even medium sized sport or sport touring motorcycles (e.g., Honda VFR) aren't cheap to run. The engines are tuned for performance not fuel efficiency and usually need premium fuel, so small cars are about as cheap to re-fuel per mile. Also the maintenance cost that killed me was tires: You have to pay around $500 dollars for (two) good tires on a motorcycle every 10k miles (if you're lucky) whereas I spent only $200 at Costco for (four) tires for my car that should last more than 50k miles.
Also consider than motorcycles depreciate faster than cars and the resale market is less liquid so you lose more money when you sell.
Summary: Buy motorcycles because they're fun, not because they're cheap!
Granted a 50cc scooter would certainly be cheaper to run than a car but then you're restricted to city streets (i.e., no highways).
Motorcycles are very much cheaper than cars.
Japanese motorcycles are totally liquid and don't depreciate quickly. You can sell a Ninja in a day.
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.