"The royals owe me. Harry owes me, Meghan owes me. What I've been through I should be rewarded for," Thomas Markle said in the documentary.
The Duke and Duchess of Sussex previously said only that they would visit Meghan's mother, Doria Ragland.
The monumental photo was the first christening portrait to feature an African-American woman.
Meghan's mother has spent the past couple of weeks helping her daughter and Harry with their newborn in the UK.
Just a week after Beyoncé was spotted shopping at a Target, the grandma-to-be of the seventh in line to the British throne was seen shopping at the budget retailer.
Royal biographer Duncan Larcombe says: "It would have been such an enormous break from royal tradition."
The Duchess of Sussex has collaborated with the Hubb Community Kitchen on a charity cookbook in support of those affected by the Grenfell Tower fire.
Meghan Markle's father, Thomas, claims he hung up on Prince Harry after a heated phone call - and claims Sunday's interview will be his last regarding the royal.