It was Bradford's first win since gaining promotion to Sky Bet League One through last season's play-offs and Parkinson said: "Mark showed whathe is all about - he is a good all-round player. We are pleased with his performance."
Yeates, who joined Bradford from Watford during the summer, quickly impressed the home fans in the 13,641 crowd as he scored the first of three goals in nine minutes midway through the first half that put the result beyond doubt.
Taking a pass from Garry Thompson on the left, he stepped inside right-back Brad Potts before beating Carlisle keeper Mark Gillespiie with a superb right-footed shot into the far corner after 20 minutes.
Last season's leading scorer, Bermudian international Nahki Wells, who had a short spell at Carlsile before joining Bradford, added the second five minutes later when he chased a headed pass from visiting skipper Paul Thirwell that was intended for Gillespie to score the second.
James Hanson scored the third in the 29th minute, taking the ball past Danny Livesey as the Carlisle defender stumbled and scoring with a low shot to give Bradford a 3-0 half-time lead.
Wells twice went close to scoring before providing skipper Gary Jones with the chance to add the fourth goal in the 66th minute, squaring the ball into his path.
Carlisle, who have conceded nine goals in their first two league matches, rarely troubled Bradford keeper Jon McLaughlin.
Parkinson added: "It is all about making a statement about what we want to do this season.
"We put people under pressure all over the pitch. We want to make Valley Parade a horrible place for visiting teams to come.
"Carlisle will be disappointed, but they are a better team than their first two league results suggest.
"The way we started was terrific. It was so important to start in a positive manner and make a statement about how we want to play, but we are keeping our feet on the ground.
"We rested a a lot of players in the midweek League Cup match at Huddersfield and there was a freshness about the team.
"We put Carlisle on the back foot from the start and the football we played in the first half was as good as we have seen here for a long time."