Pari-mutuel betting is a state-regulated industry, unlike other professional sports which are ultimately private. The state approves its existence and controls its implementation. At the top, you have state sponsored gaming boards enforcing overarching rules and legislation for gaming. Then you have state racing boards dealing with horse racing-specific regulations. You need state-employed officials to enforce said regulations on site at the racetracks.
While the racetracks themselves are privately owned and usually fully fund their day-to-day operating expenses, state money can be involved as needed. State funds may be needed to subsidize purse structure or incentives. Some states have used tax money to bail out racetracks when their budgets ran into the red.
The gaming industry brings in a lot of money for the state. Sometimes this money stays within state gaming funds, other times gets used for different sectors that don’t generate income. It’s part of the assets a state holds.