Based on the scene you posted, Owen and Beru absolutely know that Luke is the son of Anakin Skywalker. He has the same last name, it's not even hidden. After Luke complains that he's "going nowhere", Owen and Beru have the following exchange:
Luke's just not a farmer, Owen. He has too much of his father in him.
That's what I'm afraid of.
An interesting point is made in the Quora discussion linked in a comment to the original post above:
It's very unlikely that either Owen or Beru knew about that. If they did know, then Beru's statement seems remarkably...ghoulish. Listen to the way she says "He has too much of his father in him." Her statement is light-hearted, flippant, and cheerful. She's almost chuckling as she says it.
It correctly states that Beru probably doesn't know the complete fate of Anakin - however, I disagree that Owen also doesn't know. His response to her is much more grim and at the same time elusive, somewhat implying that he knows more than he's letting on. It's quite possible that Obi-Wan told him everything, but to never tell anyone else, including Beru. However, this is fully speculative.
Bear in mind, they originally met Anakin in the prequel era (Episode II, to be precise) and Anakin's mother was Owen's stepmother before she was abducted by the Tusken Raiders. It's entirely possible that she told them all about how he was being trained as a Jedi, etc. Then to have him show up in full Jedi robes only to blast off into the desert and slaughter a tribe of Tusken Raiders, bury his mother, then leave again on another adventure probably didn't make a great first impression on either Owen or Beru. If this is all Beru knows about Anakin, her flippant tone in the exchange above is justified. He comes off as a reckless adventurer with a slight dark side (remember, they all seem consider the Tusken Raiders to be beneath them).
However, while we can surmise that they both know Luke's parentage, and we can infer that Beru does not know the full story, we must concede that we just don't know how much Owen knows about Anakin and Vader. Any further discussion would be pure speculation.