There are two questions here:
Should you give your dad money for his dental work?
Can you trust your dad that he isn't lying about needing dental work.
You have to answer the first question first, because if you aren't comfortable giving him money at all, then it's a moot point. Don't rush to answer this one either, giving him $2000 for emergency dental work isn't going to solve his dental problems long term. What treatment is he saying he needs? There's no such thing as an emergency implant or denture.
So if $2000 won't solve his dental problems, then are you prepared to be the bank machine for every dental emergency that comes up? What about other emergencies? Just how much are you willing to subsidize him moving forward?
If this is the first ask, you need to contemplate this very carefully, because if he is being legit, and you pay for his needs, you will be the very first person he calls for the next need.
Okay, onto the second question.
*IF*, and it's a big if, you decide you are willing to shell out thousands for your father's emergency care, then there's the matter of determining if this is emergency care.
Well, in this case it's simple. Ask for the estimate for the dental treatment. The exact expected cost should be clear, it will also explain what the treatment is exactly.
Also, you seem to assume that you would be able to tell if he had dental work done? Why do you assume that? What treatment is he saying he's getting done that you would actually see? Most emergency treatment is extractions or root canals.
Also, definitely contact your family members to see if he's asked them for money, and to formulate a plan for how you, as a family, want to approach supporting him in the event of legit emergencies.
If you do start down the path of subsidizing his care, then your family members should at least be aware and decide if/when they want to contribute as well.
He doesn't get to set the terms of financial gifts. He can ask for his privacy OR a gift, but he can't demand both.