If a “friend” caused a part to fail in my car and refuses to pay, how can I get reimbursed for the labor and parts?

You could sue your friend for negligence.  Negligence is the failure to exercise due care (that degree of care that in this case a reasonable driver would have exercised under the same or similar circumstances to prevent foreseeable harm).  In order to prove negligence, you will need to prove duty (of due care mentioned above), breach of duty (failure to exercise due care), actual cause, proximate cause, and damages.

Actual cause means but for your friend driving your car would the clutch have been damaged?  If the answer is no, which appears to be the case, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening acts which would relieve your friend of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking in your lawsuit for negligence.  Your damages would be the cost of repairs to your car. 

You can file your lawsuit in Small Claims Court.  In addition to the cost of repairs to your car, your damages should also include court costs which would be the court filing fee and process server fee.  Use a private process server to serve your friend with the summons and complaint.  You can find process servers listed under attorney services in the Yellow Pages or online.  Don't serve your friend by certified mail return receipt because your friend won't pick up the mail and nothing will be accomplished.  Don't use the marshal to serve your friend because the marshal will only make a limited number of attempts at service and if unsuccessful will keep your money and nothing will have been accomplished.

If you obtain a court judgment against your friend, you can enforce it by getting a wage garnishment. 

The Small Claims Court advisor can help you complete the forms for filing a lawsuit and subsequently filing a wage garnishment. 

I doubt very seriously that your friend will surface to voluntarily pay you anything.  I hate to say it, but money is thicker than friendship.  I see it everyday in my legal practice.

Unless you can prove that he was "negligent" in causing the damages to your vehicle, your likelihood of winning any legal case is pretty small.  However, if you can get him to document that he previously agreed to pay for part of the repair costs, then that might be an enforceable contract.  It doesn't have to be anything formal: a text, email, letter, note (maybe even a credible witness who heard him agree).