Put the nuts in a small dry skillet and cook over medium heat until lightly toasted, about 3 minutes; transfer to a plate to cool.
Combine the nuts, basil, parsley, garlic and 1/4 teaspoon salt in a food processor and pulse until very finely ground. With the motor running, gradually add the olive oil and process until well combined.
Use a Y-shaped vegetable peeler to slice the vegetables into ribbons. With the yellow squash and zucchini, it is easiest to slice the sides, working around the core (and then discarding the core). Toss the vegetable ribbons together in a bowl. Add just enough pesto to lightly coat, then add the lemon juice and Parmesan and toss to combine. Serve immediately (any remaining pesto can be kept refrigerated for up to 1 week).