An HOA is usually made up of volunteers who own condos in the building themselves.The easiest way to understand HOA fees is that every HOA is a small business.

Or does the equity disappear and the true gift is getting rid of a problem property?

Because then you could always do Gift Funds to your daughter, for some/all of the funds needed to close the loan.

I'll be honest, it seems like it might be easiest just to cut your losses at this point. 

Is this difference a gift to her?

You could partner with a friend your age with similar goals/values to put together funds to afford a more expensive property, find a mentor and pay them a percentage on a short term loan until you can finance out of the FHA loan, or get a gift from your parents for a down payment (not a mortgage lender but parents can give $15k each tax free and then there is a 5% gift tax anything over that). 

I would be a just a little more blunt: going into a deal without sufficient capital to take into account the unexpected is the easiest way to fail.

Easiest and cheapest way.

Your parents signing over their house to you will be considered a gift, will require the filing of a gift tax return (probably not trigger any gift tax), and will reduce your parents lifetime gift exclusion. 

Perhaps the best/easiest thing is to rent and all this is just a thought experiment, but I'm curious to hear people's opinions.