Using the side of a knife, mash garlic and a pinch of salt to a paste on a cutting board. Transfer to a small bowl and add egg yolk and mustard; mix well. Whisking constantly, add ⅓ cup oil, drop by drop at first, then ⅛-teaspoonful at a time as mixture emulsifies and thickens.
*Do not add oil too quickly or aioli will break and oil will separate. Stir in lemon juice; season aioli with salt and pepper.
Prepare a grill at medium-high heat. Place sardines on a plate and drizzle the remaining 1 Tbsp. oil over; season with salt and pepper.
Grill until browned and cooked through, about 2 minutes per side. Transfer sardines to a platter and top with parsley. Serve with aioli.
Prep Ahead: Aioli can be made 1 day ahead. Cover and chill.