The son had at least one paddle, for at 6:52 he says he paddled to shore.
He does say that. I think he meant he swam to shore with the canoe. His comments in a reply on YouTube says his dad had the paddles.
Call me cynical, but do I believe what he says in the video at the time or what he says in response to a probing comment a month or so later?
I visualize that they put in above the rapid with the canoe facing upstream, essentially front ferrying out into the current. They were probably sitting instead of more stable kneeling, probably with top-heavy gear above the gunwales, probably without a downstream J-heel of the hull, and probably with no third paddle affixed to the canoe. In that visualization, the current window-shaded them upstream as soon as the bow hit the swift water.
If the son truly had no paddle, that does change the rescue options. Nothing is impossible, but it seems very unusual to me for an upstream bow paddler to end up with both paddles in a swim. It's also elementary canoe safety to have an extra paddle lashed into the canoe with some sort of semi-quick release via a cord tie, Velcro or bungee attachment, or a simple wedge.
GoPro cameras are so wide angle that they distort distance perspectives badly, but the father's island doesn't look that far from the shore. Maybe he was old and out of shape, but a reasonably fit person, already wet, could have swum the short distance from the island to the shore, especially if a rescue rope were involved. A paddle may have been throwable from the island to the son.
But if the son didn't have the skill or confidence to ferry across class 1 current or to eddy out behind an island in it, then paddle availability was moot.