No. He's playing well now, but its a Catch 22. Start him more often and his play most likely suffers. Keep the same rotation and you harm the guy you just signed.
Quick will probably never see the playoffs in LA again, no matter how well he plays. Peterson likely needs a good long run of starts to regain his form and confidence. Giving the guy with no future more starts out of respect during another losing season is detrimental to the team's future, and that is what needs to be the concern here, not honoring the past.
Maybe Peterson isn't the guy, but they just committed to him, there are no other heir apparents. Its counterproductive to sit him and not give him every opportunity to show he can pull himself together.
Quick, and the other vets, might give you the best chance to win the team's next game. The kids give you the better chance to win the team's next Cup.