"Both parties"?

Generally, there's an incumbent, and a likely winner. If you have enough popular support to win a fair election, you can probably figure out how to mobilize a few volunteers to oversee ballot counting.

I think the general assumption that you have is that all voters have equal access to financial gain, free time, and other things that in any society known are heavily deviated towards the rich. Add to that the fact that the most heavily monied campaigns normally turn out to be the winners, and it becomes very difficult to claim the idea of fair elections has any real merit except as a nice theory we want to believe is true.

I'd wager there's only really two parties with enough volunteers to actually put election monitors in every precinct

You shouldn't presume the U.S. two-party system to be the normal case.

Also as a party you don't need to monitor every precinct. You monitor those where you suspect fraud might happen. Then every voter should be able to monitor the counting process if they desire. Why should they have to rely on parties to do it for them?

Pretty sure paid contractors are not necessary, citizen will usually count for free.