People transfer property, including fractional interests, by deed. The only other common way property is transferred is by inheritance.
Occasionally, title to property passes by adverse possession, and payment of all the taxes for five years is a factor in creating ownership by adverse possession. Unfortunately, the possession of a part owner is not adverse, at least not usually, since the possession of a part owner is a matter of right. An exception is when the co-owner in possession has "ousted" the other co-owner(s) from possession. A real estate lawyer could tell you, after an interview, whether your dad has ousted the convict.
An unhappy co-owner of property can usually terminate the co-ownership by bringing a special kind of lawsuit against the other owners, asking for "partition" of the property. The court will likely order the property sold and the net proceeds of sale divided fairly. A partition may not be the best approach if all you want to do is buy out the other 1/4 and continue to stay in the property. Nevertheless, sometimes filing a suit for partition is what it takes to get the other co-owner to come to the bargaining table and negotiate a buy-out.
Finally, I might add, how is obtaining the other 1/4 going to help with finances? The 1/4 owner isn't likely to just hand over a 1/4 interest in valuable real estate! You'll probably have to pay him 1/4 of the appraised value, or whatever he thinks he can demand from you father, to get him to agree to be bought out. Make him a fair offer and see what happens. If the property is owned free and clear now, maybe a lender will assist with buy-out funding as well as some extra cash for dad.
