No. TD permits study for eligible dependants but does not provide employment authorization.
Assess the dependant’s activity in its own right
TD is a derivative status with one hard limit: a dependant admitted in TD may enrol in study, but TD carries no employment authorization and no application converts it into one. That limit attaches to the activity, not to the payer. Soliciting U.S.
clients, invoicing, handling client correspondence, running a booking calendar or doing back-office administration are work in substance, and performing them from a U.S. address for a Canadian company does not remove them from the analysis. Who may hold TD is equally fixed: a spouse and unmarried children under 21.
A common-law or conjugal partner is not covered and needs an independent route, and a child's TD ends at 21 whatever the household has planned, so the next status should be arranged well before that birthday. Citizenship and permanent residence stay separate facts for dependants too, since a spouse holding a Canadian permanent resident card but not citizenship cannot obtain TN in their own right whatever their profession. Each family member's activity determines their own route.
The principal's status still rests on citizenship in Canada or Mexico, a listed profession, that profession's stated qualification, a prearranged temporary engagement and no self-employment for the U.S. entity, and the dependants' status ends with his. Confirm a specific family situation with a U.S.-licensed immigration lawyer.