TD status, available to an eligible spouse or unmarried child under 21, does not itself authorize employment. A spouse who wants to work in the United States needs a separate basis or authorization of their own, regardless of the principal applicant's TN status.
Treat the spouse's plans as their own case
This includes a spouse who intends to keep working remotely for a Canadian or other non-U.S. employer while physically present in the United States; the employer's location does not decide whether that activity is authorized. Children's schooling plans and any age-related questions also deserve individual attention. State plainly whether the household intends to arrive together or in stages, since that affects which documents each person needs and when.
TD exists so that a professional's household is not separated by the assignment, and it is generous about presence: the dependant is admitted for the same period, may come and go, and may enrol in school or a college program. Obtain specific advice before any employment, remote activity or unpaid placement rather than assuming the academic records create permission.