Cook the pasta as directed, add the peas in for the last minute or two of cooking. Drain and toss into a bowl with the olive oil. You could skip the oil here, the sauce probably has enough. Cool to room temperature.
In the bowl of a food processor, puree the pesto, spinach, and lemon juice. Add the mayonnaise and puree. Add the pesto mixture to the cooled pasta, add the sauce a little at a time and stir you may not need the entire amount, and then add Parmesan, pine nuts, salt, and pepper. Mix well, season to taste, and serve at room temperature. I actually like mine a bit warmer than room temperature. I don't let the pasta cool all the way to room temp.