Parr began competing on the ISU Junior Grand Prix series in 2004.

Following the 2009 European Championships, Parr was named as the British entry for the 2009 World Championships, replacing Elliot Hilton, but Hilton won a lawsuit against the National Ice Skating Association and was restored to the team.

The Nebelhorn Trophy in September 2013 was the final qualification opportunity for countries which had not yet earned spots in individual Olympic figure skating events.

Parr was selected to compete at the 2014 Winter Olympics, held in February in Sochi, Russia, due to the United Kingdom's qualification for the team figure skating event.

On the first day of the Olympics, he placed ninth out of ten skaters in the men's short program, earning two points for the UK.