No. TD status carries no work authorization, and working from inside the United States for an Ontario employer is still work performed in the United States. The spouse may study and the children may attend school, but paid employment of any kind requires a status of her own that permits it.
Remote does not mean elsewhere
Hypothetical example: Yannick is admitted in TN and his wife Pilar plans to keep her salaried Ontario job by logging in from their new American address. It feels like the continuation of something already lawful, and it is the single most common misunderstanding in a TD household. Work is located where the person doing it sits, so once Pilar is living in the United States she is working there, whoever pays her and in whatever currency.
TD authorizes residence and study, not employment of any kind, and that includes self-employment and unpaid work that would ordinarily be paid. Two other facts belong in the same conversation. TD is available to a legally recognized spouse; a common-law partner is not a spouse for this purpose and needs a separate route.
And a child ceases to qualify at twenty-one, so a household with a nineteen-year-old should map that birthday against the length of the assignment now. Where the second income is essential, compare TD with the L-2 and E-2 spouse rules, under which a spouse is employment-authorized incident to status.