Following the links for visa application from the website of any U.S. consulate in India takes you to USTravelDocs.com, which appears to be the contractor handling the processing. I would thus take this site to be authoritative, and yes, your parents can appoint someone else to pick up their passports for them.
The Passport/Visa Collection Locations and Timings page provides a table of the documents required to pick up a passport; of interest to you is the second column under “Pick-up Information”:
Representative collecting passport — even in case of family members:
- A photocopy of applicant's government-issued photo ID
- Representative's original government-issued photo ID and 1 photocopy;
- Original authorization letter
In other words, if they were to ask a friend to pick up their passports for them, the friend would need to carry
- the friend's own ID and photocopy of the friend's ID
- a photocopy of each parent's ID
- a letter signed by each of the parents naming the the friend and authorizing him or her to accept the passports on their behalf. It does not specify a language, but it would not hurt to have a version in English.
Additionally, it looks like the passports need not be collected at the consulate or embassy itself. Under “Locations and Timings” there a link showing 28 additional locations which Stanley and Blue Dart will deliver to, although it seems you must have specified the delivery location in the original application.