There's a lot to unpack here so I'll only hit on a couple of points. Glossing over the details, I think your plan is sound: Buy a primary residence for your folks (it's their money, after all), buy something to house-hack, and then start acquiring properties. On the other hand, real estate is expensive right now and the timing isn't great. I'm not saying there's gonna be a crash or anything, just that relative to incomes and other metrics, real estate is historically expensive right now and thus, it might not be the best time to buy. But who knows, maybe hyper-inflation is right around the corner and it's our last chance to buy. I'm just playing the odds.
That said, if you execute your plan diligently and don't try to do everything at once, you are effectively dollar-cost-averaging into the market. Buy your folks' place. That will probably take a few months. Then when that transaction is complete, find your house-hack which I would expect to take a bit longer. By then it will be about a year from now. The craziness of the last lear will be in the rear-view and the dust will be settling from all the money that's been thrown around in the response and it will be a good time to evaluate the market for potential long-term investing. (Not going to speculate where the market might be then, since my crystal ball is broken).
The last thing I'll say is, if you read the book, then you might remember the part where the author says "plant as many trees as you can water". I'd change that to, "plant as many trees as you can water in a draught". Don't overextend yourself. If your parents are going out of the country for half the year, do they still have income? Can your income cover the PITI on 2, 3, 4+ houses? Or if there's a replay of 2020/21 and tenants stop paying rent and you can't get them out, are you going to need a bailout? If not, you might want to earmark a big chunk of that 400k for cash reserves. Just some things to think about. Best of luck to you.