Harsha, I understood your point but couldn't grasp why you're comparing the formulas when i am not using the time range anywhere. The answer lies somewhere in between what you and I are saying. Since I have introduced Quater totals in the model (doesn't matter I do not use the time range which has the quarters or not), I have introduced an intermediate level in the hierarchy between month and year, which will be consistent across the model.
That is my best guess unless someone has a better understanding
This is an excerpt from the Time Range Anaplan training ... "Levels of aggregation that are enabled in a line item, but which aren’t already included in the model, are added to the model when the time range is applied to the line item. Be aware that this could cause a Time Period formatted line item, when using the PARENT function, to return a different result than expected. For example, the formula Name(PARENT(item(time))) for Jan 18 will return FY 18 if quarter aggregations are not enabled and if quarter aggregations are enabled the result is Qtr1FY18."
I believe it explains the behavior you are experiencing.
As mentioned about time range, as you used Fy20 time range with Quarter total as dimension, thereby a part of the model time superset, hence its aggregation i.e Quarter is identified as the immediate parent for time-formatted line item , hence even in model timescale if you didnt use aggregation, due to this feature for time formatted period, anaplan is taking immediate ancestor to be Quarter and not year, thereby breaking granularity.