Corey La Barrie's parents claim "Ink Master" star Daniel Silva is responsible for the wrongful death of their son -- and they want him to pay up in court.

The YouTuber's parents, Simon La Barrie and Lissa Burton, filed suit against the tattoo artist ... alleging he and his company, Daniel Silva Tattoos, were negligent in the hours and moments leading up to Corey's untimely demise earlier this month, on his birthday no less.

In the docs, obtained by TMZ, Corey's parents allege that Daniel was driving at an unsafe and unreasonable speed when he was in the car with Corey, which they say means he was operating the vehicle without due care and in violation of California law.

Not only that, but they go on to allege he was under the influence at the time of the accident -- which ultimately killed Corey -- and they even go so far as to say that his own company provided the intoxicants, which the La Barries say was alcohol and/or other unknown substances. Then the nail in the coffin ... they say Daniel had been popped before for driving under the influence, so he should've known the dangers of doing so.

As we reported ... Daniel has been charged with murder in connection to Corey's death. Now, he's got a civil suit to deal with too.

Corey's parents say that, with the loss of their son, they're going to be missing out on a whole lot of emotional comfort he did and would've provided them for years to come -- not to mention the burial expenses they've incurred and other costs.

For that, they want a judge to grant them damages. We've reached out to Daniel's attorney for comment ... so far, no word back.