  We study conformal higher spin (CHS) fields on constant curvature
backgrounds. By employing parent formulation technique in combination with
tractor description of GJMS operators we find a manifestly factorized form of
the CHS wave operators for symmetric fields of arbitrary integer spin $s$ and
gauge invariance of arbitrary order $t\leq s$. In the case of the usual
Fradkin-Tseytlin fields $t=1$ this gives a systematic derivation of the
factorization formulas known in the literature while for $t>1$ the explicit
formulas were not known. We also relate the gauge invariance of the CHS fields
to the partially-fixed gauge invariance of the factors and show that the
factors can be identified with (partially gauge-fixed) wave operators for
(partially)-massless or special massive fields. As a byproduct, we establish a
detailed relationship with the tractor approach and, in particular, derive the
tractor form of the CHS equations and gauge symmetries.
