As far as I know you are right, Carol.
Worming is also done for show reasons. It does look nice to see a light coloured rope wormed with dark string.
Parceling was done before sticky tape was invented, in those times old, worn out sails where always at hand on (sailing) boats. These sails could be cut (or better ripped) in narrow sections. This would be used as tape. Sticky cloth tape will do the job and might well be easier to use.
Serving is winding string round the rope. It has to be done very tight and the different rounds should touch each other so water has no way to get in between the different layers of the protection.
There are several tools to do this job but the one that amazes me most is one that looks like a mallet with a bite missing. The missing bite is where it rests on the rope. The string is wound round the handle and kept under tension. The mallet with the string is wound round the rope. Sometimes it has a string holder, otherwise it is a job for a man and a boy (as they used to say.) In the old serving mallets the handle is severly bitten into by the string.
If you have acces to a copy of Ashley Book Of Knots, there is a picture on page 511, (Between ABOK 3082 and #3083 and some more info on pages 539/540, (#3336/#3351).
The final part of the job would have been waterproofing. I think tar was used but maybe other things too.
And if I used different words from what you did, maybe my spelling/words are wrong, I did look up the ABOK numbers, not the spelling.
Willeke