Chris,
Page 8 section 47of Part 97 defines "Third party communications" as:
A message from the control operator (first party) of an amateur station to another amateur station control operator (second party) on behalf of another person (third party).
There is no first party.
There is no third party.
There is no "on behalf of" aspect.
The definition of "Third party communications" does not seem to be met here.