Yes, green card parents can get health insurance. In fact, there are a few different ways for green card holders to get health insurance coverage.
One option is to enroll in a health insurance plan through the Marketplace. The Marketplace is a government-run website where you can compare and shop for health insurance plans. If you qualify for a subsidy, you may be able to get help paying for your monthly premiums.
Another option is to get health insurance through an employer. If your employer offers health insurance, you may be able to enroll in a plan through them. If you're not sure whether or not your employer offers health insurance, you can contact them and ask.
You can also purchase a private health insurance plan on your own. This can be done through the Marketplace or directly from an insurance company. If you're healthy and don't have any pre-existing conditions, you may be able to get a less expensive health insurance plan.
Finally, if you're a low-income green card holder, you may be eligible for Medicaid. Medicaid is a government-run health insurance program for low-income individuals and families. Each state has different eligibility requirements, so you'll need to check with your state's Medicaid office to see if you qualify.