Merry Father's Day, Dad!
Why is "Happy Father's Day" incorrect?
I clicked through the link you provided and cannot figure out what the message is trying to convey. It seems British at first, given the use of "Mum" instead of "Mom," but one of the children is named "Train".....?? In any case, Father's Day originated in the US (in 1910), and the traditional greeting has always been "Happy Father's Day." As emsr said, the only holiday greeting that ever uses "Merry" is Christmas.