As a general rule, even if the card itself does not specify it, it authorizes residents to work in Spain, under the same conditions as a Spanish.
There is a trick to consider:
what if the relative is in charge, like your case?
When the regrouping is an ascendant or a child over 21 years of age, or in cases where there is an economic dependency, they must maintain this condition of "being in charge" to maintain the card. However, this condition will not be affected if it proves that it carries out a labor activity in which the income obtained does not have the character of resource necessary for the sustenance.
What happens if I stop being in charge? If the relative ceases to be in charge, he would no longer be entitled to this community card, so he should change his situation to the general scheme requesting the authorization that corresponds in his case (employed, self-employed…).
Check: Tarjeta de residencia de familiar de ciudadano de la Unión (Spanish)