My understanding is yes because of what I've read on the BA site:
BUT I just wanted to confirm that this is indeed true... you know, just to calm my mindCan my family members work in the UK?
A family member of an EEA national who intends to travel with them or join them in the UK can work without a work permit. Family members are generally not required to register under the Worker Registration Scheme or comply with the special requirements for Bulgarian and Romanian workers. The EEA national must be a 'qualified person' in the UK.