No. TD dependants may live in the United States and may study, but they hold no employment authorisation. A partner who wants to work must qualify independently.
TD is not a work permit for anybody
Dependants admitted in TD status are permitted to accompany the principal and to enrol in study, and they appear among the classes expressly allowed to study. Employment is not included and there is no application that adds it. Households where both adults work should understand this before the principal accepts a hospital offer, because the classification chosen for one adult effectively decides the other's working life for the duration.
There are workable answers: the partner may qualify for TN in their own right if their profession is listed and they have their own American employer, or an employer may sponsor them in a classification that permits work. One further point specific to unmarried couples: derivative status depends on a legally valid marriage, so a common-law partner is not a TD dependant at all and would need an independent basis to enter and remain.