How long do my parents have to stay in USA before they can leave without losing their green card?

That is not really the right question. They are lawful PERMANENT residents. So the intent is that they will live in the US. Regardless of the 180/365 rule, DHS really is looking at a pattern of practice. How much time in a year are they in the US? Where is their employment and house? Where do they pay taxes? I recommend that you set an appointment for them with a competent immigration attorney in your area once in the US. They need to go over their fact pattern and long term travel goals so the attorney can outline the law and see what options are available to protect them. Make sure to ask about the reentry permit.

There is no specific length of time during which a recent green card holder must remain in US before traveling overseas. They can come in today and return to at any time. You must however note that if a green card holder travels overseas, it is advised that they do not remain outside the US for more than 180 days. Should they do so, it raises a rebuttable presumption that they wish to abandon their residence [green card] That presumption may be overcome. But once a green card holder leaves the US and remains overseas for a pcontinuous period of 365 days, the green card is gone

There is no requirement of how they long have to stay once they come in as immigrants. However, they cannot be outside the country for more than a year at a time. If they plan to live abroad for more than a year I would encourage them to get a travel permit which allows them to stay outside the country for 2 years.

In order for your parents to retain their residency and eligibility for citizenship, your parents cannot stay outside of the US for more than 6 months each year. However, they can travel as soon as they receive their green cards.