I'd agree on the Alice/Jasper but for this poll I chose Esme and Carlisle.
As to why:
Over the centuries, Carlisle has learned self-control over his instinct and taught his adoptive children to adapt with him. He, I think, is a very genuine person, honest and loyal. Someone to confide in no matter what and someone who will listen quietly no matter the situation and wait until all the details are retrieved before making a verdict. He has taught the others to be loyal and accepted each Cullen from the start, trying to save each from their deaths, and give them a new life. Thus his vampiric trait of compassion.
Esme lost her child and wanted death. Carlisle saved her and, when he saved Edward, Rosalie and Emmett, she gained three adoptive children. And with Alice and Jasper, she welcomed them into the home also. Carlisle and Esme's relationship has had a long time to develope and that love clearly hasn't wavered over time. Both Esme and Carlisle are generous to Jacob and the Pack, allowing them to stay at their home and such. They welcomed Bella into the family and Renesmee Carlie. They make excellent grandparents for Nessie.
I think the best reason I chose Carlisle/Esme as my favourite non-Bella Edward couple is that they accept each other and their children's (and grandchild's) faults and strengths and are there to support and listen to each other like a husband and wife should.