The new Birds Barbershop location on East Sixth Street will include free parking for customers. (Courtesy Birds Barbershop) Birds Barbershop’s location on East Sixth Street is relocating to a new building two blocks west with validated parking for customers. The location at 1107 E. 6th St., Austin, opened in 2008 offering haircuts and color for men, women and children. Birds will found at 1401 E. 6th, St., Austin, beginning March 1.