A divorced Father who wants no direct involvement with his child and whines about having to provide financial support, yet plasters his Facebook page with pictures of his son to give the impression of being a loving Father.
Dick is such a Facebook Father, he spends all his child support on gambling and alcohol and never speaks to his son, yet he gets a tattoo of the boy and posts pics on FB.
Man whose sole contribution to the upbringing of his child consists of Facebook postings extolling his virtues as a father. A Facebook Father wants all the rights of fatherhood with none of the responsibilities and uses Facebook to fool the causal observer into believing that he is Father of the Year.
See also: Deadtweet Dad
That Facebook Father just posted a beautiful picture of his daughter, but hasn't paid child support, medical expenses or any tuition for over a year.