The "public money" is the pool of voluntary checkoffs on income tax. It's not from "general fund" or FICA or some other tax/fee source.
Again, why is the federal government involved in payoffs to parties? The first priority for these funds is the nominating conventions. If people want to support their parties, they can do it directly. I just don't like that the parties have so ingrained themselves in the government that they might as well be the government. Washington would not be happy with the current state of affairs.
BTW, for party-related expenses directly taken from the taxpayers, just look at all the perks and extra staff given to minority and majority leaders and whips in Congress when the only positions actually authorized are the Speaker of the House and the President pro tempore of the Senate. The parties should be paying for that extra staff and expenses, not us.