Second year apprentices Jack Stockdill, Jack Bentley and Nathan Curtis have also been told that they will not be receiving professional contracts and they too will also depart the club.
Fellow apprentices Oliver McBurnie and Niall Heaton will, however, move up into Parkinson's senior squad for next season.
McBurnie signed a professional contract with the football club back in August while Heaton was also offered a pro deal with City as part of the move that brought him back to Valley Parade from Liverpool just under a year ago.
Thompson leaves the Bantams having made 102 appearances for City, including starts in the 2013 Capital One Cup Final and League 2 Play-Off Final last season.
Having scored 11 times for the club in total, probably Thompson's most famous moment in a City shirt came when he memorably opened the scoring in the Bantams' Capital One Cup Quarter-Final with Arsenal in December 2012.
Thompson's final goal for Bradford was the match-winner in the club's last home game of the season against Crawley Town in late April.
Former Middlesbrough defender Bates joined City back in October, making his debut at Crawley back in October. He would go onto play in 22 further games for Parkinson's men during the this season
After signing his first pro contract last summer, Swain unfortunately failed to feature in any of City's League 1 fixtures during the 2013/2014 campaign.
Discussing his thoughts on the three departing senior players, Parkinson said: "I would really like to thank Thommo (Garry Thompson) for his significant contribution over the two years he has been here at the club.
"He played his part in helping us to a major cup final and promotion last season, and he also contributed to a mid-table finish for us this season.
"I am sure there will be no shortage of clubs interested in his services.
"I'd also like to thank Batesy (Matthew Bates) for his professionalism and hard-work throughout the season. With regards Louie, I've spoken to him and I just feel he needs to find a club more suited to his level so that he can benefit from getting regular games under his belt."
Parkinson also confirmed that discussions with other members of his current squad whose contracts are due to expire in the summer will take place from the end of this week onwards.
Some of these talks may be affected by Parkinson's exact playing budget for next season - a figure which will be finalised when the City chief speaks again with the club's chairmen later this week.
Parkinson added: "Discussions with the board continue to take place regarding the budget for next season.
"We plan to speak with some of the other players at the end of this week and the beginning of the next one. The discussions with those players will be based on the result of the talks I have with Julian (Rhodes)."