You need to bond the subpanel ground bus to the plumbing
You will need to run a bonding conductor from the subpanel ground bus to a grounding clamp on the plumbing, as per NEC 250.104(A)(3):
(3) Multiple Buildings or Structures Supplied by a Feeder(s) or
Branch Circuit(s). The metal water piping system(s) installed
in or attached to a building or structure shall be bonded to any
of the following:
(1) Building or structure disconnecting means enclosure
where located at the building or structure
(2) Equipment grounding conductor run with the supply
(3) One or more grounding electrodes used
The bonding jumper(s) shall be sized in accordance with
Table 250.102(C)(1), based on the size of the feeder or branch-circuit conductors that supply the building or structure. The
bonding jumper shall not be required to be larger than the
largest ungrounded feeder or branch-circuit conductor supplying the building or structure.
This ensures that if the plumbing somehow became energized, the breaker would trip and disconnect the fault, instead of leaving the plumbing floating off to shocking voltages.
Don't worry about parallel paths here
As long as you have one and only one neutral-ground bond, located at the main service entrance, you are safe from parallel-path problems. Since ground wires don't normally carry current, they are free to run in parallel with each other, metal plumbing systems, metallic structural elements, and so on.