In a bowl, mix the canned salmon, breadcrumbs, eggs, mayonnaise, Dijon mustard, celery, red onion, fresh parsley, salt, and black pepper until evenly combined, with no dry crumbs left. Stir until the mixture looks cohesive and clumps when pressed (no visible eggy streaks).

Form the mixture into 8 equal patties about 3/4-inch thick. Set them on a plate and refrigerate for 15 minutes so they hold their shape and don’t crumble when flipped.

Stir together mayonnaise, sweet relish, lemon juice, salt, and black pepper for the tartar sauce until smooth and creamy. Refrigerate for 15 minutes while the patties chill so the flavors meld.

Heat butter and olive oil in a skillet over medium heat until the butter foams lightly. Place patties in the skillet and cook for 5–6 minutes, until the bottoms are deep golden brown and you can see crisp edges.

Flip patties and cook another 5–6 minutes until both sides are deep golden brown and the patties feel set in the center. Remove to a plate and let them rest briefly so the interior stays moist.

Serve salmon patties hot with tartar sauce alongside lemon wedges and fresh dill garnish. Spoon sauce over or at the side and cut one open to confirm the center is moist and herb-flecked.