Disclaimer: I'm Asian. My dad is currently staying with us in our 2 bedroom apt, helping us care for our 1 year old, and doing our chores for free for 3 months--I'm so so grateful. This is NOT what most people want to do in their retirement in Western cultures. My mom did the same for us for 10 months, but is now caring for my grandparents.
We have other relatives visiting this week, and my dad is sleeping in the mostly un-finished basement (not our choice--he refused to go any other way). My parents paid full college tuition for me and my siblings. They were going through some cash flow issues when we bought our first property, but made us a big loan for the down payment on the second--the first became a full investment property and we repaid them very quickly. They are incredibly proud of me, and would not have loaned us the money if they didn't think we were decent person. My dad is now offering to pay for renovations to build out a patio for us--he thinks it will increase the value of our property, and it actually might.
Although my parents are FI and do not expect me to support them financially for their retirement, I know that I would not hesitate to provide anything for them if they ever needed it. I have a much easier relationship with my parents than some of my friends who are expected to be their parents retirement fund. My husband's father has also made us some generous gifts, and though we never needed them, we've just rolled the gifts into our financial plan; and if it ever became a sticking point, we could easily transfer the money back and say, no thank you, not if there are any strings attached.
I guess my point here is that I have no issues with gifts from close family. About #3, is it possible that your parents don't realize that you are being intentional about the small house or perhaps worried that you are picking a small place because of the cost? They might be saying, we'll give you the money, don't feel bad accepting it because we'll get to stay there when we come and visit. And don't worry about us, we have money in the bank for our retirement (though off-topic it depends on their age and SS on whether they have enough).
One more thing... all the Asian parents I know seem to think that property is a very good investment, especially in cities. I don't fully get it, but they all seem to be encouraging their kids to get property early in life for future capital appreciation, and not thinking as much about taxes/mortgage/maintenance... they might be thinking small gift now, grows to a bigger gift later.
If your parents are like anything like mine, they would much rather give you cash than buy your plane ticket to go overseas, give you hundred dollar gifts at the holidays, or send you a gift card in the mail.
So yeah, that's me not stereotyping Asian parents at all ;)