I am an Indian citizen living in the US for last 4 yrs. If my parents are visiting me for 6 months in the US, can I apply for Obamacare to support their health.
Generally, persons visiting or vacationing in the US legally can use the Marketplace to shop for health insurance, but aren't eligible for cost assistance and aren't required to have health insurance. However, if the person is required to file taxes, or is a qualifying tax-dependent while in the US, they are generally subject to the law and all of it's provisions for the tax year. This means they would qualify for cost assistance, and be responsible for the fee for not having coverage (or for obtaining an exemption).
TIP: Obamacare is not a health insurance. It is a slang term for the law (The Patient Protection and Affordable Care Act) which regulates private and, to a certain extent, public health coverage like Medicaid and Medicare.
Short answer: Anyone who is in the US legally can use the Marketplaces to shop for health insurance. However, only legal residents and their tax-dependents can qualify for cost assistance or qualify for public health insurance programs. If your parents are only coming to visit for 6 months, meaning are not working or earning an income (not on a work visa) and don't qualify as tax-dependents (see qualifying relative sections here) then the best option may be traveler's insurance or short-term health insurance. Catastrophic coverage can also be a good option for those who are generally healthy, but want to know the costs would be capped in the event of a major medical "catastrophe". That being said, I would still start by using the marketplaces to compare different private insurers prices and coverage side by side.
If your parents do qualify as your tax-dependents in your household, than you would be responsible for ensuring that they are covered with health insurance (via a family plan) for the ENTIRE YEAR. You may qualify for cost assistance through the Marketplace, but you'd also owe the fee for not having insurance for each month they went without insurance during the year or get an exemption.
The states are given some leeway to cover more individuals then are required by the Affordable Care Acts. So, you'll want to check with your states marketplace or healthcare.gov to be certain of your options.