Dependents in TD status are tied to the principal's TN status remaining valid and consistent. 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.
Keep the household's status picture stable
Confirm dependents' documents remain consistent with the first employer's terms while the second authorization is being arranged, since that is the relationship their status is currently tied to. If the added income or schedule change affects household planning, such as childcare or a spouse's own work arrangements, discuss that separately from the immigration authorization questions, since the two issues have different evidence and different decision points.
TD status is available to a qualifying spouse and children, but it does not authorize employment; calling services occasional, remote, unpaid, or consulting does not settle whether they constitute US work. The first review should produce a household plan that names the principal route and every separate work, study, or travel question.; each conclusion should point to the record that settles it. Dependants hold TD status, which authorizes residence and study but no employment of any kind, and working remotely from inside the United States for a Canadian employer is still work performed there.