After further research, I now believe that Peter might have been another relative of Tom Day. They were both in Collin County, Texas at the same time and Peter Day moved to Somervell County, Texas very near where Tom Day and his wife, Margaret Jane Meek Day lived on the Paluxy River. At the time he moved there Somervell County was part of Hood County.
I have not been able to make a direct connection with Tom Day's family and Peter Day. He maybe a cousin or Uncle of Tom Day's father.
Hope this helps.