Some states they do. In my state, the Democrats hold caucuses instead of primaries (even though the state runs primaries anyway—weird story), and I think at the

Some states they do. In my state, the Democrats hold caucuses instead of primaries (even though the state runs primaries anyway—weird story), and I think at their own expense. Running an election is a pretty big operation (you need to register voters, check that they're eligible, set up and staff either polling places or mail-in sites, and so on. The state already has most of that infrastructure, and it seems wasteful to duplicate it. I don't know for sure, but I wouldn't be surprised if some (or all) primary states allow third parties to register to have their primaries run as well.