Clarke, 26, penned a two-year deal with the football club at City's Apperley Bridge training ground this Friday afternoon to commit his future to the Bantams until at least June 2016.
The Cork-born attacker becomes Parkinson's fifth signing of the summer only days before the club are due to return for pre-season training.
Clarke will officially join City on Tuesday (1 July) when his contract with previous club Crawley Town expires.
The Irishman is able to play in a number of attacking positions, including left wing, striker or in the 'number ten' role just behind the main forward.
Like fellow recent signing Alan Sheehan, Clarke is also comfortable taking free-kicks and penalties with his cultured left foot.
Clarke said: "I have to reiterate really what most new signings say when they arrive here. This is a massive club and the fan base here is very big.
"Those factors, coupled with the chats I had with the manager before I came here, made this the only place I wanted to come to and join.
"Everyone who plays at Bradford always mentions the size of the crowd and the size of the stadium. That sort of thing really does stick with opposition players when they come up to Bradford.
"When you look at the manager we have here and the quality of the players he has brought in already this summer, this place really did tick all the boxes for me.
"I like to be creative on the ball and make things happen for other players and myself. That is why I feel I will fit in with the way the manager is wanting to play his football next season.
"I'm exciting to start pre-season next week with the boys and get started. I'm delighted to sign here."
Parkinson added: "I am pleased we have managed to bring Billy to the football club. It is important that we add more competition in attacking areas, and Billy is another player I know who is relishing the challenge of playing for Bradford City."
Clarke has previous experience of playing in League 1 with Crawley but he has also spent a fair chunk of his earlier career with Championship clubs Ipswich Town and Blackpool.