It may well be unlawful age discrimination to refuse to promote you because your association with your elderly relatives is causing your employer to treat you less favourably than it would other workers.
However, claims like this are complicated and it is a good idea to talk it through with your union rep if you have one, or by using the services of Acas, to find out whether the issue can be resolved, perhaps through in-work mediation, where a trained outsider attempts to broker a compromise between the two parties. In-work mediation is voluntary and confidential. Acas is the TUC’s preferred mediator. You can find out more about mediation on the Acas website.