You don’t see them so much any more, but in yesteryear when “the interstate” was still a new thing, nearly all entrance ramps had signs that explicitly stated (to the driver attempting the merge) to “Yield Right of Way” or simply “Yield”. Some ramps still have that sign, of course, but they’re not as prevalent as they once were.
On normal streets with driveways and parking lots and right-angled side streets with stop signs and so forth, the presumption is “whoever is first” has right of way. So you can feel confident (legally) in pulling out of a driveway as long as the road traffic has the opportunity to slow and even stop if necessary. You were there first, so you have right of way. (Obviously, it’s not always wise to drive this way!) But that right of way does not apply to interstate highways and other high speed roads which are designed and operated so the drivers do not “normally” have to be on the lookout for merging traffic.
Your mother is correct. It is up to the merging driver to “make a safe merge”, and not the responsibility of the driver on the main road to “make way”, to move over or slow down. (Of course, it’s often the prudent thing to do and nearly always the polite thing to do, but that’s for smart and courteous drivers to figure out for themselves. Given modern traffic patterns and frequently crowded conditions in urban driving, this isn’t as absolutely true as it is on rural highways.) The onus is always on the driver attempting to join the flow of traffic, and the signs are there (or were there) to remind him of that fact. (I think it’s one of the basic rules of the road explicitly pointed out in driver training manuals issued by every state DMV; you don’t have to take my word for it.)
In point of fact it is always “safe to slow down” when conditions warrant that. Obviously your father wanted to merge “at speed” into the highway traffic, and not join the traffic from a standing start, which might require the high speed drivers to make emergency maneuvers to avoid him. That’s part of the “yield” that he has to make. If for some reason he is forced to make a start onto the road from a standing start or a very slow speed, then he has to wait for that opening in the traffic, in order to make a safe entry. Now, once he’s on the road, and presumably accelerating up to highway speed, if drivers coming up behind him are forced to slow because of his current lack of speed, but if they can do that safely and in a non-emergency manner, then that’s just normal driving.
The thing is, “not joining the road from a standing start”, which your dad might have had to do if he had been forced to stop at the mouth of the entrance ramp (some busy highways, such as around Los Angeles and other highly congested areas actually have red and green traffic lights at the mouth of the entrance ramp!) is for his convenience. Having to wait for a larger gap in traffic would have inconvenienced him, and he didn’t want to wait and then have to judge an abnormal (accelerate from the stopped position) entry onto the highway.
But that’s what was called for. Either slowing to pass behind the (possibly distracted, confused or simply inattentive or discourteous) driver who was already in or partially in that lane, or pull forward with enough additional speed as to make the overlap disappear – but that may have been unsafe as well.
He should have slowed on the entrance ramp – or even stopped – and merged with a better gap.