I’m not sure what model parent thinks will work. Maybe rent control light where price increases are limited but the limit is highish? I can see that working out, limiting increases to 1.5-2X inflation might keep the market fluid and provide some stability to renters without locking new tenants out.
My parents have seen a 4x increase in the value of their house in 20 years, is that unfair?
If we wanted to make the market fair, we would disallow home owners all together. This effectively happens for most people in some places in Europe. Also, I heard of some places in Europe where if you own your own home, you have to pay some extra tax vs. what you would have paid in rent. I think Switzerland? They also had some form of rent price control, although this just made it really hard to rent an apartment even if you were willing to pay more. These are fairly complex systems that all give advantages to existing residents (so moving to Europe for two years to do a post doc is much harder than it would be if you did that in the USA).