All of the payments at closing happen at closing. The bank puts its money into the closing attorney's escrow account, the attorney then distributes the money to parties owed.
In this case the seller doesn't leaving with 100% of the purchase price... he forgoes being paid the last $1.2 million (the exact amount the bank wouldn't lend you), but gets a 2nd mortgage note signed by you instead. This leaves everyone happy. The bank get's a first mortgage with only lending their LTV. The seller gets his purchase price in cash, less the 2nd mortgage amount that he lends you, and you get the park with $0 of your own funds down.
The bank and especially the seller will take some convincing of this. You have to inspire confidence in your ability to make the Park work and run in a financially viable way, such that that the sellers don't mind being one of your investors. Pay attention to the balloon payment specifics on the 2nd mortgage. If the seller gives you the typical 5 year balloon payment, make sure you have a real plan to pay off the note or refinance after increasing the value of the park in 3 or 4 years. You don't want to be caught when the 2nd mortgage goes away in 5 years with no refinancing alternatives.