Any flights operated by a certificate holder must be operated in accordance with the authorization of their certificate. Authorization will vary.
An Air Carrier may conduct training1 flights under §91 if authorized to do so. If not authorized, any flights operated by that Air Carrier—including training flights—must be conducted under the part of 14 CFR (§121, §135, etc) specified by their Air Operating Certificate (AOC).
An Air Carrier may be authorized to conduct training flights under 14 CFR 91 in accordance with their Operation Specifications authorized by the FAA.
The exception to this would be if that Air Carrier's Operation Specifications do not authorize the Air Carrier to conduct flights under §91. I know of no reason for such exception, nor if any such exception has been withheld from any actual Air Carrier.
The authorization is found within the Operation Specifications of the company in question, not necessarily in 14 CFR, though it can be inferred there. See Operation Specification A001 for authorization to conduct training flights under 14 CFR 91, for example:
d. The certificate holder is authorized to conduct flights under 14 CFR Part 91 for crewmember training, maintenance tests, ferrying, re-positioning, and the carriage of company officials using the applicable authorizations in these operations specifications, without obtaining a Letter of Authorization, provided the flights are not conducted for compensation or hire and no charge of any kind is made for the conduct of the flights.
(The above is an example only; actual authorization may vary.)
An Air Carrier may also be authorized by A031 of their Ops Spec to make arrangements with an outside party for Contract Training, including instruction and evaluation. An authorized outside party under this specification may include a §142 Training Center and/or a different §119 Certificate holder authorized to conduct training and/or checking. Any training flights operated by either such certificate holder must likewise be authorized by the terms of their certificate.
1Note: Regarding the question of what is included under the definition of the term training—particularly whether checkrides are included in that definition—no explicit guidance is offered. However, a definition might well be inferred from—among other things—the definitions for the following two terms as found in Ops Spec A002, Definitions and Abbreviations:
Major Contract Training
Any flight training, flight testing, or flight checking leading to and maintaining certification and qualification of air carrier flight crew members in accordance with the requirements (maneuvers and procedures) explicitly stated in 14 CFR Parts 61, 121, or 135; or in SFAR 58 Advanced Qualification Program (AQP), as applicable.
Any training, testing, or checking activity which an air carrier certificate holder provides by way of a contract arrangement with another party.