Botham's mother, father and sister spoke out for this PSA -- produced by Roc Nation and the NFL -- explaining why the world "lost a great man" the night Amber Guyger walked into his apartment and shot him.

In the video, Bertram and Allison Jean say their son was "destined for greatness," and was everything to them -- but his father adds, "Life is not sweet anymore."

Their heartwrenching story is the latest project from the "Responsibility Program" -- launched by Jay-Z's Roc Nation. It's produced multiple PSAs, highlighting victims of police brutality -- but this is the first one sponsored by the NFL.

As for why Botham's folks are speaking out for the PSA -- his mom says, "What I hope to see happening is that our black boys are not seen as a threat." His sister, Allisa Findley, adds, "The color of my skin can be perceived as a weapon, and it's not."

You'll recall the tearful courtroom scene last October when Guyger was sentenced to 10 years in prison. Botham's brother shockingly said he forgave Guyger, before hugging her.

Botham's parents reference that extraordinary moment, saying there is no need for hatred.

Similarly, Michelle Kenney -- whose son, Antwon Rose Jr., was shot by a Pittsburgh police officer -- told TMZ ... she hoped the PSAs would "open eyes and bring them into our world," so America can have "more compassion" when these tragedies happen.

And, Angella and Danroy Henry told us their PSA effectively showed the world who their son, DJ Henry, REALLY was ... beyond the news headlines about the Pace University student-athlete killed in 2010 by a police officer.

We're told the NFL will run a social justice themed PSA -- about ex-WR star Anquan Boldin's murdered cousin -- during halftime of the Super Bowl.

That spot is part of the league's non-profit initiative, Inspire Change.