Early to mid June is not quite peak season. It might be harder to find a spot in Jasper, but as long as your RV is not huge (under 20 ft), I think you should be able to find a spot. You would just need to do without a hook up - those spots have long since booked up. I would use the online system, making sure you are looking for all campsites that meet your requirements (you can do without a hook up - there will be water taps and sani dumps at the big campgrounds).
There are a number of first-come, first serve campgrounds - often nicer and in more scenic places then the big reservable campgrounds. Should be fine if you are coming mid week or can arrive by midday on a Friday.
If all the campgrounds are full -- and that only generally happens on holiday weekends and maybe one or two other weekends -- Parks Canada will open overflow campsites. These are essentially gravel parking lots - no facilities other than outhouses. But some can be surprisingly scenic. At least one in Jasper, and one or two in Banff NP. I don't think they've ever been open in June, which suggests that campgrounds haven't ever been near or totally booked in June (other than the very end near Canada Day).
If the overflow lots are full, then you have to leave the park to find a place to stay. When that happens, they generally will post signs by the park entrances. I've never heard of that happening other than on long weekens -and you aren't coming anytime close to a long weekend. There are no other places for overnight RV parking in the parks -- and even daytime RV parking can be tight.
They are very, very strict about illegal camping. I strongly doubt the RVs you saw had been at Moraine Lake overnight - they most definitely patrol that parking lot and being caught will result in a nasty fine. The only reason they would have been legally there overnight is if the occupants were staying overnight at a hut or bivy -- you then are allowed to park overnight (but have to register your vehicle when you get your bivy permit/hut reservation). The van may have gotten away with it in 2014, but these days, no way, no how. Same for overnight parking anywhere in the townsites.
From Jasper, you'd be looking at heading back into Hinton - there are provincial campgrounds in the area, plus at least one grocery store that allows RVs in their lots. No overnight parking on residential streets or in other commercial lots. From Banff, your first option would be Canmore -- things may change this year if towns/cities want to discourage RV camping outside designated campgrounds, but RVs or vans have been allowed to stay overnight in the info centre parking lot and also in one area of the parking lot by the Canadian Tire (up by the railroad tracks). Beyond that, you're headed east to where there are provincial and private campgrounds.