Scrub mussels, pull off any tough "beards" from sides of shells, and discard any mussels that aren't tightly closed. In a large pot with a tight-fitting lid, bring wine and onion to a boil over high heat. Add mussels; cook, covered, until they open, 2 to 3 minutes. Uncover and let stand until cool enough to handle. Discard any closed mussels.
Separate mussels from shells using a sharp knife. Set each mussel on a half-shell (choose flattish ones) and place on a rimmed baking sheet. Cover and chill.
Toast saffron in a small frying pan over medium heat, stirring, until slightly darker, 1 1/2 to 2 minutes.
In a blender, whirl garlic, roasted pepper, saffron, cayenne, and lemon zest and juice until smooth. Add mayonnaise and whirl to blend. Scrape into a bowl.
Pour salt onto 2 platters. Set mussels level on salt. Dollop mayo on each and top with a parsley leaf.
Make ahead: Through step 4, up to 1 day, chilled airtight; through step 5, up to 3 more hours, chilled uncovered.