The parking brake would just lock the ends of the axles. It shouldn't play any part in if there is any movement of the carrier due to worn bearings. As an idea, you could try having someone turn a wheel while you watch the carrier move and then repeat on the other side to see if anything does anything but turn.
I also echo making sure there is on pieces in the axle tubes. While you are at it, I might be a good idea to drop the drive shaft and check for any play in the pinion bearings as a preventative maintenance measure. It won't have any bearing on fixing your issue but based on the look of the residual oil in your axle, a fluid change was long over due and the pinion bearings could be worn as a result. Better to find out now instead of putting it all back together and find out you have other issues.