How much work is it reasonable for an object constructor to do? Should it simply initialize fields and not actually perform any operations on data, or is it okay to have it
A constructor should set up the object to be used.
So whatever that is. That may include taking action on some data or just setting fields. It will change from each class.
In the case you are speaking of an Html Parser, I would opt for creating the class, and then calling a Parse Html method. The reason for this is it gives you a furture opportunity to set items in the class for parsing the Html.