Preheat the oven to 400 degrees F. Prick each yam with a fork in a few spots and arrange on a baking sheet. Bake until soft, about 45 minutes.
Meanwhile, heat the olive oil, grapeseed oil, whole garlic cloves and saffron in a small saucepan over medium-low heat until the garlic is soft, about 15 minutes. Remove from the heat and let cool completely. Remove the cooked garlic with a slotted spoon and transfer to a cutting board (reserve the infused oil). Using the flat side of a chef's knife, mash the garlic with a pinch of salt until smooth; set aside.
Make the aioli: Put the egg yolks, lemon juice, 1 tablespoon water and 1/2 teaspoon salt in a medium heatproof bowl. Bring a few inches of water to a bare simmer in a medium saucepan, then turn off the heat. Set the bowl with the egg yolks over the saucepan (do not let the bowl touch the water); whisk just until the egg mixture thickens slightly, 30 seconds to 1 minute, then remove the bowl from the saucepan. Using a hand mixer or a whisk, gradually beat in the saffron-infused oil in a slow, steady stream until the aioli is thick and creamy. (If it gets too thick, beat in 1 to 2 teaspoons water.) Mix in the mashed cooked garlic, grated fresh garlic and 1/4 teaspoon salt. Cover and refrigerate.
Remove the yams from the oven and let cool slightly. Split open and fluff the insides with a fork. Thin the aioli with water, if desired, and spoon onto the yams. Sprinkle with parsley.