This is a minor problem, since public services make up a very small portion of government spending in the US. Most government spending is merely redistribution.

http://usgovernmentspending.com/piechart_2009_US_total

Public services would be provided by businesses like everything else. "Law and order" would be maintained primarily the way it is now, which is through social custom. But since there would still be aggressors, you would have to pay for private security. And you would probably want to carry a weapon with you.

All of these issues are discussed extensively in the libertarian literature. Murray Rothbard and Stefan Molyneux are good places to start.

Perhaps public services (including the legal system) could be funded by donations instead of tax money.

Your parent probably hasn't thought it out that far; your parent probably started at "gee, I don't like paying taxes" and stopped at "therefore, taxes are bad! Down with taxes!"

Alternatively, your parent could belong to the set of people who basically figure the money will just come from somewhere. They don't really care where, so long as it's someone else's money.