Parker launched the new service following a request from the council and Coventry City Football Club.
The new service will run late into the night to "support the Arena's ambition to stage more concerts and events", Parker said.
The deal will also allow for extra 'football special' buses on busy matchdays.
Parker added: "This is about making it much easier and quicker for thousands of football fans to get to the game when taking the train to and from Coventry.
"Football is such a massive part of people's lives, and that's why we're investing millions to improve matchday travel for fans."
Councillor Lynnette Kelly, cabinet member for jobs, regeneration and climate change, said it is "great news for travelling fans and local people".
"It's set to be an exciting season on and off the pitch at the CBS, and the club and the wider city are ready to welcome fans from across the country for games back in the top-flight," she added.