There is no "maximum period of time a resident is allowed to be outside the United States in order not to lose the Green Card". Abandonment of residence is determined by looking at many factors, not just time outside the US. It is a subjective determination that also looks at her purpose for being abroad and whether it had a definitive end date, whether she was unable to return due to reasons beyond her control, whether she retained access to a home in the US, whether she had family that remained in the US, whether she filed taxes in the US, etc. It's hard to say whether she would be deemed to have abandoned residence in her situation; on one hand, her parent being sick is a reason why she stayed away for longer than expected; however, it's been years and it seems that her absence is open-ended and not really temporary.
If her card is still valid, she could fly to the US with it and simply try to enter. Although the green card is not an acceptable document for re-entry after an absence of one year, the officer has discretion to waive that requirement and let her in anyway if the officer determines that she has not abandoned residence. If the officer believes she has abandoned residence and asks her to sign I-407 to voluntarily abandon residence, she should refuse. In that case, she will be temporarily let in and given a notice to appear at removal proceedings in immigration court (because only an immigration judge has the power to involuntarily revoke her permanent residence). If it goes to immigration court, and the immigration judge agrees that she has not abandoned residence, the removal proceedings will be closed and she will continue as a permanent resident.
The alternative method is to apply for an SB-1 returning resident visa at a US consulate abroad. However, those require a very good reason that she wasn't able to return earlier due to circumstances beyond her control, and are rarely granted. The chances of success this way are much worse than going to the US and presenting herself to the immigration officer or immigration judge.
If she has abandoned residence, you can of course just petition her to immigrate (i.e. become a permanent resident) again from scratch.
