"The barber is giving a haircut to my father" sounds unnatural in this case, because "giving ... to" normally means that a specific THING is being given by one person TO another person, but the barber isn't actually GIVING him a THING. " You would only say "a NEW HAIRCUT" if he was getting his hair cut in a different style from what he normally has.
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shravanmmHI,I am getting my hair cut.Yes, that's good.