Are you required to record flight hours of serialized parts?
Short answer: only for the airframe, engines, propellors and rotors (as applicable).

This legal interpretation is a response to someone who asked which records are required to identify when life-limited parts need to be replaced. It says that you're only required to keep flight time records of the airframe, engine(s), propellor(s) and rotor(s), as appropriate:


  Existing regulations, specifically 14 C.F.R. § 91.417(a)(2)(i), require
  each owner or operator to keep records containing the total time in
  service of the airframe, each engine, each propeller, and each rotor.
  This is accomplished by the owner or operator recording and tracking
  in some form and manner the time in service of the airframe,
  engines(s), propeller(s), and rotor(s) from the moment the aircraft
  leaves the surface of the earth until it touches it at the next point
  of landing, as referenced in 14 C.F.R. § 1.1. In addition, §
  91.417(a)(2)(ii), and similar provisions in 14 C.F.R. §§ 121.380(a)(2)(iii) and 135.439(a)(2)(ii), require owners or operators (certificate holders) to keep records that show the current status of
  life-limited parts of each airframe, engine, propeller, rotor, and
  appliance.


91.417 does indeed say - among other things - that owners and operators have to record the time in service and status for the airframe, engines, props and rotors, but only the status for life-limited "appliances":


  (2) Records containing the following information:
  
  (i) The total time in service of the airframe, each engine, each
  propeller, and each rotor.
  
  (ii) The current status of life-limited parts of each airframe,
  engine, propeller, rotor, and appliance.


1.1. defines "time in service":


  Time in service, with respect to maintenance time records, means the
  time from the moment an aircraft leaves the surface of the earth until
  it touches it at the next point of landing.


And 43.10 defines "life-limited parts":


  Life-limited part means any part for which a mandatory replacement
  limit is specified in the type design, the Instructions for Continued
  Airworthiness, or the maintenance manual.


To summarize:


You only need to record service time for the airframe, engine, props and rotors
For any other life-limited parts, you only need to record their status, i.e. you need to be able to say when it's time to replace them