What does "parallel in-register" mean for a protein structure in a fibril?
There is a clear explanation and figure showing this in the following reference:

Anton Gorkovskiy, Kent R. Thurber, Robert Tycko, Reed B. Wickner, PNAS 2014, 111 (43) E4615-E4622.

A parallel beta sheet is one where the direction from N- to C-termini on adjacent strands run in a parallel direction (rather than antiparallel). In-register means that each residue in one molecule is aligned with the same residue in the neighboring molecule, producing lines of identical residues along the long axis of the filament.