Merriam-Webster has an entry for half-orphan, meaning someone with only one living parent. They say specifically a child, which would match the usual usage of "orphan" (as mentioned in the question), but it might be used of an adult sometimes, either jocularly or by extension. This doesn't distinguish which parent is dead, but could be combined with a further explanation if it matters.
New York Post: I’m a ‘Disney Adult’ — I tried ‘Disneybounding’ to fight costume ban
I’m a ‘Disney Adult’ — I tried ‘Disneybounding’ to fight costume ban
Fox News: ‘Disney Adults’ anger some online after they invent ‘Disneybounding’ to avoid costume ban: 'My personal hell'
‘Disney Adults’ anger some online after they invent ‘Disneybounding’ to avoid costume ban: 'My personal hell'