Doran Martell never took a paramour even after Mellario left him and he was the ruling prince of dorne. Oberyn might've had a paramour but he never married and he got into a duel with another dornish lord because he slept with that lord's paramour. Myriah and Maron never took any lovers or paramours. Just because the culture might accept something doesn't automatically mean everyone there is going to do it. So don't stereotype. And by the way Elia would NEVER have been allowed a paramour regardless of whether such a thing is acceptable in her homeland because it certainly wouldn't be acceptable in Kings landing and she would be risking her and her children's lives for doing such a thing.
So Doran never took a paramour because he married for love, not politics. Even if that romance eventually fizzled out, he was already an older father with gout by then. Oberyn's duel with another lord over a cheating paramour emphasizes the fact that polyamory isn't cheating, but actual cheating is still reviled in Dornish culture. And yeah, it's very likely Elia didn't have a paramour, especially not publicly. But she lived in Dragonstone, not King's Landing. She would travel to court from time to time, and Aerys brought her there during Robert's Rebellion, but she would have spent much more time at her and her husband's castle. But people do dumb things for love, and having multiple dornish Kingsguard and ladies in waiting, it wouldn't be hard to keep "rumors" away from Aerys.
Elia and Rhaegar married for duty, had children for duty, and they seemed to be good partners is parenting their children. But maybe they both found love elsewhere, or maybe not. At the end of the day, we don't know. People say Rhaegar loved Lyanna, but we don't know what name he said as he died on the Trident. We don't know if Lyanna was kidnapped, or if she went willingly. We don't know if Elia knew Rhaegars plans, or if he abandoned her unawares. All we have are possibly false visions, and the eternally unreliable memories of those who knew them.
Maybe my view of Elia, Lyanna, and Rhaegar is optimistic. Maybe Elia was a sickly victim in all of this. Maybe Lyanna was a hypocrite who was fine being the other woman, but not being cheated on. Maybe Rhaegar cared more about his interpretation of philosophy than he did the people he had to hurt to achieve it. But maybe Elia was scheming right alongside Rhaegar. Maybe Lyanna understood that love shared openly was different than secret trists. And maybe Rhaegar really loved his northern lady, as well as his family
