Is ATC clearance or permission required to fly helicopter routes and transitions in the Los Angeles area?
According to the FAA's JO7210.3AA Facility Operation and Administration (chapter 11, section 5):


  Pilot adherence to charted helicopter routes and the recommended
  altitudes or flight ceilings associated with them will normally be
  voluntary. However, controllers may assign charted routes and
  altitudes and expect or request pilot compliance with them, provided
  such procedures are called for in specific FAA−operator Letters of
  Agreement, or are necessitated by traffic density and/or safety
  considerations; controllers also may restrict operations within
  designated operating zones when requested by local law enforcement
  officials and the restriction would not adversely affect other
  aircraft operations.


In other words, it's mostly a voluntary thing but pilots should still be aware of them and prepared to follow them if instructed.

And you have to consider the airspace as well: large parts of some routes are in uncontrolled class G airspace, e.g. the PALISADES coastal route, west of Santa Monica. ATC doesn't control traffic in class G, so asking for permission wouldn't make sense.