Ok, well I guess I'm going to be the different opinion in this thread...
I think pedigree is very important. Especially nowadays.
I think your horse's pedigree is ok. Sire is stronger than the dam, but I like that the dam traces to Lady Bugs Moon. There's a lot of "run" on the dam's side. I just wish everything was pushed forward a couple generations. Streakin La Jolla was fast and produced a lot of great horses. He passed on his speed and ability. The hitch is, that while there is plenty of speed, there needs to be the proper conformation to allow all that speed to scrunch up and nail a turn.
I find it kinda neat... I, myself, have a horse that traces to Yesteryear x Bar Beau Bea. Achieve To Recieve Quarter Horse
Judging by your horses pedigree alone, he should have enough speed to barrel race. Now, the missing pieces of the puzzle are the horses true ability, conformation, and heart. If your horse has all of those qualities, then with a great trainer, yes, your horse can be a successful barrel horse.