Connor played for Norwich's Under-13 in Euro Cup at Heiden (Germany) in January 2013 and then played for theUnder-14s in tournament in Woudenberg. near Utrecht in June 2014

He signed as a scholar in July 2017 before earning his first professional contract in the summer of 2019.

He left the club by mutual consent on 2nd December 2019 having made three appearances for the Under-23s in 2019/20. He then joined Norwich United.

On 10th September 2020 it was reported that Kettering Town manager Paul Cox was keen to bring Parsons to the Poppies. He also trained with a number of National League sides during the 2020 summer pre-season. On 1st October 2020 it was reported that Parsons had joined Lowestoft on a season long loan. Blues boss Jamie Godbold told the club’s website: “It’s great to welcome Connor to our club. He has shown ambition throughout the summer to return to the highest possible level and that drive will be much needed for him and us to be as successful as we want to be. I am convinced if he can show the work ethic and ability I have seen so far he will be a fans favourite and if the opportunities present themselves to him, we will do what we always do and help them fulfil their ambitions”.

On 15th March 2021 Connor moved to Iceland to join UMF Tindastoll who play in the Icelandic third division.

On 24th August 2021 he signed a one year contract with Wycombe Wanderers. Wycombe's Development team coach Sam Grace said: “Connor is a really talented player who has impressed us during his trial period with the club, and provides extra competition for places on the left-hand side. We’re pleased with how the development squad is progressing and we’ve already seen the emergence of Andre Burley and Olly Pendlebury into the first-team this season, and hopefully Connor can follow in their footsteps and continue to kick on in his career."

He made his professional debut for Wycombe on 31st August 2021 in the Football League Trophy against Aston Villa's Under-21s, scoring in that game. He played twice more in that competition before joining Notts County on a two month loan on 26th November 2021. Parsons played three games for the Magpies.

On 3rd February 2022 he was loaned to National League side Bromley on a one month deal.

On 17th August 2022 he was loaned to National League side Solihull Moors until 3rd January 2022 to build up his first team experience. He made 10 appearances for the Moors.

On 23rd December 2022 he signed for Irish side Waterford for an undisclosed fee. Parsons said: “I am delighted to have got the deal over the line and to have signed with Waterford FC. It is a fresh start for me and I am really excited to be here. I have spoken with Danny on a number of occasions and it seems to be a really exciting time for the club and I am happy to be a part of that. Our goal will be to win the league and get the club promoted. I have done some research and I know that Waterford should be playing at the highest level here. The club has done really well to retain the core of last season’s squad, so the mentality from last year coupled with some new players will be exciting. I can’t wait to get started. I think the fans will find me to be an exciting player. I like to move the ball with speed and take risks. As footballers we are there to entertain and that is what I want to do. I want to get the fans up on their feet.”