Cook pasta according to the package instructions; drain and set aside.
Bring a small saucepan of water to a boil; add the peas and cook until softened, then drain.
Grate Parmesan (if needed), slice garlic, juice lemon, and chop basil.
Add the peas, Parmesan, garlic, 2 tablespoons lemon juice, basil, and pepper in a blender or food processor; mix until smooth, slowly adding the olive oil as it blends.
Serve pesto over cooked pasta with strawberries on the side.