No. TD status allows residence and study but carries no work authorization, and part-time retail work is employment. He would need his own category, such as a TN in a listed profession if his own qualifications support one.
TD is a residence status, not a work status
The spouse and unmarried children under twenty-one of a TN professional may be admitted in TD status. That status lets them live in the United States for the same period as the principal and lets them attend school or college, but it does not authorize any employment, paid or in kind, and a store job is employment however few the hours. A spouse who wants to work has to qualify independently: if he holds a degree in a listed profession and a U.S.
employer offers him a professional position, he can apply for TN on his own facts; otherwise the household should plan on one income during the TN period. A TD spouse who is not a Canadian citizen needs a TD visa from a U.S. consulate before travelling, whereas a Canadian spouse can apply at the port of entry with the marriage certificate and the principal's TN documents.
Children age out of TD at twenty-one regardless of study.