Who pays for primary elections? Taxpayers. Who are primaries for? The private political parties. Who can participate in primaries? Only people of a party (“closed primary”). Some states allow “open primaries.” Independent/unaffiliated/nonpartisan voters cannot vote in most primaries.
Can you vote for candidates of two or more different parties in a primary? No. Is this fair or lawful? No.
The political parties should pay for their own elections.