FCL.305 CPL — Privileges and conditions
(a) Privileges. The privileges of the holder of a CPL are, within the appropriate aircraft category, to:
(1) exercise all the privileges of the holder of an LAPL and a PPL;
(2) act as PIC or co-pilot of any aircraft engaged in operations other than commercial air
transport;
(3) act as PIC in commercial air transport of any single-pilot aircraft subject to the restrictions
specified in FCL.060 and in this Subpart;
(4) act as co-pilot in commercial air transport subject to the restrictions specified in FCL.060
FCL.505 ATPL — Privileges
(a) The privileges of the holder of an ATPL are, within the appropriate aircraft category, to:
(1) exercise all the privileges of the holder of an LAPL, a PPL and a CPL;
(2) act as PIC of aircraft engaged in commercial air transport.
Does this answer your question?
The key word here is "Commercial Air Transport".