Thanks a lot for the suggestion Dan. I have some surprising news from what tried.
Lets call the nii I created from PAR-REC, my.nii and the nii file I got from the scanner, scanner.nii

When I do simple 3dTshift on the my.nii, I still get the same warning that the data is already aligned in time! Since you suggested that PAR/REC is native, and 3dPAR2AFNI -n must not be doing time-correction either, its surprising that 3dTshift gives this warning on my.nii.

The 3dTshift command I gave was
3dTshift -tzero 0 -quintic -prefix whatever my.nii

I tried what you suggested. Here's what happens:

When I convert my.nii to my.BRIK using 3dTcat, I don't get any warning and the size of the data file (my.BRIK) does not change.
When I convert scanner.nii to scanner.BRIK using 3dTcat, I get warning about conversion of datatype from INT16 to FLOAT32 and the size of the data file doubles. So, scanner.BRIK is twice the size of my.BRIK

Then, when I do diff my.BRIK scanner.BRIK, the output just says that the two differ, not surprizing since the datatypes are different.

When I use 3dcalc to subtract my.BRIK from scanner.BRIK, I get warning about grid mismatch!!

Looking at my.BRIK in AFNI shows values from 400 to 800 (approx), while scanner.BRIK shows values from 800 to 1100 (approx)
Looking at my-scanner.BRIK in AFNI, I see values in hundreds, as expected.

What do you think is happening?
Thanks a lot again,
Gaurav