I don’t know if there is a difference between a “sponsored green card“ and a regular green card, but generally a green card means that a person is a permanent US resident. Your parents would also be US residence. If they pass the substantial presence test, meaning they lived in the US at least 183 days over the past three years, counted using a special formula.
https://www.irs.gov/individuals/international-taxpayers/substantial-presence-test
To claim your parents as a dependent, they must be US residents, and you must have provided more than half their total financial support for the entire year, and they can’t have more than $4400 of taxable income each.
You said they don’t have income “in the US.” If your parents have green cards and are US permanent residents, even if they are not living in the US at the time, their entire worldwide income is subject to US taxation. (That’s one of the prices for the benefit of having US residency.) Also, since their entire worldwide income counts as US taxable income, you can’t claim them as dependents unless their entire worldwide income is less than $4400 each.
As a special note, understand that if your parents have US residency, but do not file US tax returns as required, that can jeopardize their immigration status, and their ability to be “promoted” from green card holders to citizens.
Although it is not part of the dependent definition that your parents must live with you for any specific period of time, I suspect that from the way, you asked the question, they do not meet the income test, or you did not provide more than half of their total support for the entire year, or both.
