There are 2 different options for you:
Option 1
Your father can sell the house and invest only the Long Term Capital Gains property in the new house. Repeat its not necessary to invest entire sale proceeds. He can invest only the Long Term Capital Gains part.
He will be the co-owner in the new house and his share can be proportionate to his investment.
Its possible for you to apply for a housing loan with both yourself and your father as co-owners.
Option 2
As your father is a senior citizen, he may gift the present house to you and or others of his choice ( if its a self acquired property for him), as part of inheritance planning. If the entire property is gifted to you, you can then sell the property and invest in the new property, which will be held in your name. If there are others, who may inherit the property, you may buy their shares or they may even continue as co-owners.
When the property is to be transferred to you, there will be stamp duty on the registration of the gift deed. This is an additional cost. However, when you sell the house, the cost in the hands of your father will be cost in your hands. Hence, there will not be significant difference in tax liability. (Variance in tax liability may be due to difference in applicable tax rates and other sources of income).
An additional issue is that as you are a resident in USA, your residential status will also be counted. The buyer needs to deduct tax u/s 195 of the Income Tax Act while making payment to you, if you are a Non Resident in the year in which you sell the house. If your father is a resident, then TDS will be applicable and that too @ 1% only, if the sale consideration is Rs 50 Lakhs or more.
So you need to weigh both the options and then take an appropriate decision.