Did you go back to live with your parents then?
You technically did not have any state residency status for the 2 months you did not maintain a home in the US. For those 2 months, any income you earned would only be federally taxed, not state or local taxed.
If you maintained a home base during those 2 months in IL (returning to your parents home), then your residency status for those 2 months would be IL.