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Antonio Callaway had gun parts and ammunition in his car (in addition to weed) when he was pulled over in Ohio ... and the police video obtained by TMZ Sports shows it made the lead officer get "real nervous."

The Cleveland Browns rookie WR -- selected in the 4th round of the 2018 NFL Draft -- was pulled over on August 5 at 2:59 AM in Strongsville, Ohio for a routine traffic violation.

The cop smelled weed and searched the car -- finding a small amount of marijuana under the driver's seat ... along with the backstrap of a Glock firearm and several rounds of ammo.

You can see the officer gets very serious when he finds the bullets and gun parts -- ordering Callaway to put his hands in the air and stay put.

Antonio told the officer there was NO gun in the car -- just the parts. The officer eventually verified Callaway's claim, but admits to his partner he got anxious when he found the bullets.

In the end, the officer realizes Antonio is a Cleveland Browns player and admitted he felt bad busting him ... since he didn't appear drunk or high and was super polite during the arrest.

At one point, the officer admits he's excited to watch Callaway play -- saying he's "fast as sh*t."