You need to make sure that the request hasnt come from your father, some carers know that some people simply need that little bit extra support and do try to do things, so if your dad wants mum to have her hair washed so she's clean and tidy and her dignity is maintained because of this, and that he cannot physically take her to the hairdressers, and if dad and the carer is happy with this and the price is reasonable then why rock the boat. especially if the carer is doing this in her time off and not during works time. Also, its a bit of different company and probably gives your dad a bit of a break while the carer is with her.
On the other hand, if dad hasn't requested it, and the carer is simply using peoples vulnerability to make money then you should contact social services and advise this is happening and not just the agency, if you dont get a quick enough response then request the team manager or put it in writing as councils have to respond within certain timescales when they receive something. Also, make sure your dad realises that if the agency is contracted via social services he pays social services and not the carer direct. If this is the case I would insist the council ensure the carer returns the money gained from dad.
Also, why isnt one of mums morning calls increased in time to give the morning carers time to wash her hair? there are even methods and equipment (speak to an ot) to wash peoples hair who are bed bound, usually the social worker would make sure that when the person has an increased call for bathing or a full strip wash, that there is an extra half hour (eve if its once a week) to wash the persons hair and dry it during the same call. ask the social worker for a reassessment to increase mums homecare times.