They are joined back at the panel and at the fan- that is a parallel conductor.
Those, not that.
310.4 Conductors in Parallel.
310.4(A) General. Aluminum, copper-clad aluminum, or copper
conductors of size 1/0 AWG and larger, comprising each
phase, polarity, neutral, or grounded circuit conductor shall
be permitted to be connected in parallel (electrically joined
at both ends).
Exception No. 1:
Conductors in sizes smaller than 1/0
AWG shall be permitted to be run in parallel to supply
control power to indicating instruments, contactors,
relays,
solenoids, and similar control devices, or for frequencies of
360 Hz and higher, provided all of the following apply:
(a) They are contained within the same raceway or
cable.
(b) The ampacity of each individual conductor is sufficient
to carry the entire load current shared by the parallel
conductors.
(c) The overcurrent protection is such that the ampacity
of each individual conductor will not be exceeded if one
or more of the parallel conductors become inadvertently
disconnected.
Pipe from the panel, to switch box 1, to switch box 2, to switch box 3, then to a relay. Use #12 on a 20a breaker.... done deal.