A extremely lazy girl that goes to the barber
By googling anime hair cut in park I got this:

Kuro to Kin no Hirakanai Kagi

The synopsis seems to match quite well with your description:


  Katagiri Kanade is a hikikomori who spent her days lazing off in her room. Her specialty is her broad imagination which often leads to various wild delusions. One night a weird cowboy guy appeared in her dream, saying that he's her heart's hero. Due to his encouragement, Kanade decided to try and go have a haircut the next day. It was then when she met a mysterious guy who cut her hair in the park, giving her courage to resume her school days.