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.
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.
Should everyone be allowed to use these public services when not everyone is donating? Should you be allowed to use the police even if you haven't been donating to their budget? What if the police have two calls: one from someone who has donated and one from someone who hasn't? Who has priority?
The tax system exists to fund services that are generally not used by someone until they are needed (FEMA, Medicare, unemployment, etc.). I can honestly say I would never pay into any of these programs unless I had to and I think most people would do the same (even if they wouldn't admit to it openly).
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.
http://usgovernmentspending.com/piechart_2009_US_total