This previously answered question addresses what is really meant by the idea that the Arthurian Merlin "lived backwards". The description does not really fit anything we know about time travel and timelines from DW mythology.
Quoting from the accepted answer, Merlin is said to "remember what is in our future", but to "have no knowledge of what is in our past". This doesn't describe either River or the Doctor. Both of them remember what is in their own personal past, as well as any historical information they have gathered along the way as they travel through space and time. The Doctor and River both have linear timelines, they just don't experience events in time in the same order as those who never travel through the time vortex.
I don't think there is any evidence that the writers of 'Battlefield' attempted to address every Athurian legend ever written and this idea of Merlin living "backwards" is not something contained in most popular stories so I doubt it would have been considered if it isn't directly addressed in the script. There is a recurring idea in Doctor Who that various legends are obscured versions of events he has had a hand in so perhaps it should be taken as given that any power attributed to Merlin in legends are distortions of The Doctor's gifts and abilities.
However, I should add that it is not even true to say that River's timeline is running "antiparallel" to The Doctor. It is a fact that the first time the Doctor met River was her last time meeting him before she died, so his future was her past. However, their subsequent meetings are apparently not in strict reverse order either, otherwise, they would not have had to "synchronise diaries" when they met.
AMY: They're both time travellers, so they never meet in the right order. They're syncing their diaries.
- The Impossible Astronaut
And of course, the last time The Doctor met her in The Husbands of River Song was not River's first. In fact, it was her penultimate meeting with The Doctor, as he gifts her with the sonic she will use just before her death.