Not on the basis of TD status. TD provides no employment authorization, including work performed in the United States for a foreign payer.
Assess the spouse as a separate prospective worker
A qualifying spouse or unmarried child under 21 may seek TD, but the principal’s unused opportunity does not become a family work benefit. The spouse needs an independently appropriate basis before performing the work.
The remote-work question comes up constantly and the conservative reading is the one to plan around: the location of the person doing the work matters, so continuing a Canadian job from a kitchen table in the United States is not a safe workaround. Some households manage the risk by having the Canadian business run day to day by an on-site manager, with the spouse holding only an ownership interest and taking distributions rather than a salary for services performed while present. Adding a second employer does not usually change dependent status directly, but any disruption to the first employer's authorization while arranging the second could indirectly affect dependents, so keep both employer relationships clearly documented and stable. A spouse counting on working should raise that with counsel early, because TD permits study but does not permit employment, and remote or unpaid activities require individual review for an outside employer.