No. There is a $250,000 exclusion of profits from the sale of a personal residence. ($500,000 if married filing jointly). You can take this exclusion multiple times as long as you have lived in the home as your principal residence for at least two years.
BUT, here’s an answer to a question you didn’t ask! If you and your brother are co-owners of the new house WITH your dad (or is your Dad just living there?) then there are some important documents you must create right now — before he becomes incapacitated.
You should go to an experienced estate planning attorney and have a Revocable Living Trust created — and then re-title the house in the name of the RLT. I will assume that he is trustee — and you and your brother will be named “successor trustees” if he cannot act. (Think stroke, or dementia.) But all three of you could be co-trustees. Someone must have the authority to make decisions, spend money (think roof repair, new furnace) and those duties and sources of funds should be clearly defined in the RLT agreement. The goal is not only to manage the house well while all three of you live there together, but to avoid fights when your father passes on! And you should open a bank account in the name of the RLT and have your Dad’s SS checks direct deposited there. I assume you two brothers are going to contribute to the cost of maintaining the house — so perhaps you should each agree to deposit a set amount to this checking account — to cover property taxes, insurance, utilities, etc. That way all will be contributing equally — and there will be no future arguments!
If my math is correct, you two brothers must be in your early 60s or younger. That means you likely have another 30 years to live in retirement. You certainly can’t plan to live off your Dad for all that time!!Do some planning for yourselves, as well.