I am no expert on cleaning of parchment, but I would strongly discourage any use of aqueous solutions if there is the slightest indication that the parchment may be soiled by water-soluble dirt or carries water-soluble ink. If this is the case, there is a strong risk that moisture will couse the dirt to spread out on the parchment and soil it irreversibly. I think there are metods to render water-soluble ink insoluble before using aqueous cleaners.
CaCl2 is very hygroscopic. That may, on the long run, make your parchment wet and destroy it. The alternative, storing in very dry air is harmfull for the parchment. So, definitely, you should get rid of the CaCl2.
While calcium forms stable salts in contact with sulfate, you may use fumes of sulfuric acid. They may convert the CaCl2 to CaSO4, like this:
CaCl2 + H20 + SO3 --> CaSo4 + 2HCl
CaSO4 is stable and solid. You may remove it mechanicaly, when necessary.
HCl will mainly evaporate by itself, but you can also neutralize it using NH3 from NH4OH solution. The NH3 may be harmfull to the parchment, so maybe you should leave this part out.
According to my informations, acid is used in production of some kind of parchment, what makes me certain, that it is not harmfull to the parchment, but I am not that certain about the colours.
You should write to or contact the Department of Histories and Methods for the Conservation of Cultural Heritage (DISMEC) which is located in Ravenna, Italy.
One subject of reserach is the Analysis of the Environment and Diagnostics of the damages inflicted on cultural heritage. They should know what to do. Contact Prof. Salvatore Lorusso (http://www.unibo.it/docenti/salvatore.lorusso).