Heat grill to high. Brush salmon on both sides with oil and season with salt and pepper. Grill on both sides until golden brown and cooked to medium doneness, about 3 minutes per side. Remove from the grill and immediately drizzle with the morel vinaigrette.
Place morels in a bowl and cover with boiling water. Let sit until soft, about 30 minutes, then drain, reserving liquid, and coarsely chop.
Whisk together vinegar, shallot, mustard and salt and pepper in a medium bowl. Slowly whisk in the olive oil until emulsified. Use a little of the morel soaking liquid to thin, if needed. Fold in the mushrooms and parsley.