My FIL paid off his home, moved out and has given it to us as our first investment property to get started. The as is appraisal came back at $375K and we have taken out a HML of $120K. Once rehab has been completed AR...

Joe gives out gifts and flowers and such to his tenants.

All of that can be a gift or a combination of gift and seller concessions. 

You can get a gift for a down payment but it must be from a  blood relative or "like" relative  and the you must explain the close relationship and why the "like" relative would give you a gift. a GIFT for downpayment is money given willingly to a buyer without repayment. 

If you are purchasing with a conventional investor loan you are not able to get gift funds or borrowed funds for the cash to close.

I looked for weeks trying to get a lender that had a manageable down payment who also accepted gift funds ( borrowing most of my down payment ).

Oh, great gift for the holidays!"

A family member gave my girlfriend and I a gift of a substantial amount of money to start a business.

My question is: Can my parents gift me some of their home equity they have built in the home they live.

That's like asking a restaurant to give you a $20 discount on your bill next time because you paid with a $20 gift card this time.