Parr was born in Oslo where he grew up in Holmlia and played for Holmlia SK. During his youth he also played for KFUM Oslo and Hauger before he got enrolled in the NTG/Lyn-program. Parr made his Tippeligaen debut for Lyn on 30 April 2006, in a 1–4 loss against Stabæk. He played 14 senior matches for Lyn, 11 of them in Tippeligaen. Parr was mostly used as a striker or winger at Lyn. In March 2007 he signed a three-year contract for Aalesund. He was in the starting line-up five times, and came off the bench 14 times during his first year in Aalesund. His first league goal came in the 4–1 victory against Sandefjord on 29 July 2007. He won the Norwegian Football Cup with Aalesund in 2009. In February 2010, he extended his contract an extra year, expiring at the end of 2011.
Crystal Palace
On 16 July 2011, Aalesund agreed a fee for Parr to transfer to London-based Championship side Crystal Palace, subject to a medical. Three days later, Crystal Palace confirmed that Parr had signed a three-year deal for an undisclosed fee. on 24 January 2012, Parr missed the final penalty when Crystal Palace lost a penalty shoot-out 1–3 against Cardiff City in the semi final of the 2011–12 League Cup. On 12 May 2012, he was voted Crystal Palace player of the year for the 2011–12 season, due to his work-ethic and his great fitness. Parr started the 2012–13 season well, and made the left-back position his own in Dougie Freedman's team. After Freedman's acrimonious departure from Selhurst Park in October, Parr's place in the starting XI was never under threat. New manager Ian Holloway had to also deploy him in the right fullback position because of injuries. Parr's season ended in April against Ipswich Town after damaging ligaments in his ankle. Nevertheless, Crystal Palace still managed to gain promotion to the Premier League at the end of the season. The Eagles beat Watford in the promotion-playoffs finals to return to the English top tier after an eight-year absence. When Parr returned from his ankle-injury, Dean Moxey was the preferred choice as Palace's left-back. Parr got his Premier League debut for Crystal Palace on 21 December, replacing left-back rival Dean Moxey at half time in the match against Newcastle United. Parr gave away a penalty in his first appearance in eight months. Parr then started the next eight matches for Palace under the new manager Tony Pulis, before he was injured in the FA Cup fourth-round match against Wigan Athletic on 25 January 2014. Palace lost the match 1–2. At the end of 2013–14 season, he was released by the club.
Ipswich Town
On 7 July 2014, Parr signed for Ipswich Town on a free transfer, signing a two-year deal. He made his debut for the club on 30 August 2014, starting in a 1–1 draw with Derby County. He scored his first goal for Ipswich on 16 September 2014, netting the opening goal in a 2–0 win against Brighton & Hove Albion at Portman Road. Parr made 33 appearances during his first season at the club, scoring twice, as Ipswich finished 6th in the Championship, qualifying for the Championship play-offs as a result. He made his first appearance of the 2015–16 season on 15 September, featuring as a second-half substitute in a 0–1 away win over Leeds United at Elland Road. He scored his first goal of the season on 24 October in a 1–1 draw away at Nottingham Forest. Parr found regular game time limited during his second season at Ipswich, making just 10 appearances in all competitions over the first half of the season, scoring once. On 13 January 2016 he moved to Strømsgodset on a permanent deal for a reported fee of around £70,000.
Strømsgodset
Parr signed a permanent deal with Strømsgodset on 13 January 2016. He signed a 3.5-year contract.
