I am a Japanese and a permanent resident. Back home, my parents saved
money under my name through my life. I moved to US in 1994 after
getting married with an American in 1993. As I probably are not going
back home for rest of my life, my parents decided to sell all the
stocks they had under my name, and it amassed to $60,000. They want to
send all that money to me here in U.S. at once. Am I taxed on this, or
do I have any obligation to report this to IRA?
They said that that is my own money saved up under my name, it would
not be any problem. Is this true. When money was in stocks, the stock
company was reporting to US treasury every year.

If you are a US resident then you need to report your income from all
sources worldwide. This means that your gains on sale should be
reported. Also, if you have foreign currency and then convert it to US
dollars, there could be an exchange gain that would also be income.

The movement of the cash itself is not an issue from a tax perspective
other than any foreign exchange gain.

It's also worth knowing that gain on stock sale is source according to
the residence of the seller. If you're a US resident, the gain would
be US source even if the stocks are traded on a foreign exchange.