"Der Vater des Jungen, mit dem du gespielt hast"
The boy's father, with whom you have played
this should be accepted as well.
It is said in the sentence "Der Vater des Jungen, mit dem du gespielt hast" that you played with the boy. Your translation has different meaning: you played with the father of that boy.
I don't know why you were downvoted, you are completely right. Bernardsia's sentence "The boy's father, with whom you have played" indicates that you played with the father, and is wrong. The German sentence is saying that you played with the boy.
- Who is that over there?
- That is the father of the boy with whom you played.
"des Jungen" is a Genitiv case from singular Junge. I found this table here http://de.wiktionary.org/wiki/Junge : Kasus: Singular - Plural (1); Nominativ: der Junge - die Jungen; Genitiv: des Jungen - der Jungen; Dativ: dem Jungen - den Jungen; Akkusativ: den Jungen - die Jungen