According to the book "the love you make" by peter brown, who was with them and mentioned in the song the ballad of john and yoko, he says it was Paul. Just citing what I read.
First I've heard of Paul playing the drum solo on Abbey Road, and I've read "The Love You Make." In the book "The Beatles Gear," Andy Babiuk says the song "The End" "features Starr playing his first and last solo on a Beatles record." Babiuk goes on to quote Iron Butterfly drummer Ron Bushy as saying that in 1971, McCartney and Starr came to see an IB show and that Ringo told Ron he had "copped" his solo from In-A-Gadda-Da-Vida" for "The End." There are similarities in the Abbey Road solo. Paul is a heck of a drummer, but I doubt he would steal Ringo's solo here.
