A spouse and unmarried children under 21 may accompany a TN professional in TD status. TD dependants may attend school, but TD carries no work authorization of any kind, and each dependant must qualify in their own right.
TD is a residence and study status, never a work status
TD exists only while the principal holds TN, and it ends when that status ends. A partner in TD could not take a local job, could not freelance, and could not continue running a Canadian business from a desk inside the United States, because work performed physically in the country is work regardless of who pays for it or where the payer is located. Children in TD may enrol in school.
Two points deserve early attention in a household planning around forestry seasons. A common-law partner is not a spouse for these purposes and would need an independent route. A child who reaches 21 loses TD eligibility on that birthday, so a family with a teenager close to that age should plan that person's own status well ahead rather than discovering the problem at a renewal.