At 2500 rep, one can work to create tag synonymous (see https://softwareengineering.stackexchange.com/help/privileges/suggest-tag-synonyms ). While I am not familiar with all the behind the scenes working, I believe it would merge the two tags so that nothing is lost.
That said, while similar, I'm not sure they are the same or should be synonymed (thats a new word I just made up). A parser is a thing - a specific implementation. While parsing is a concept or a process that the parser does.
One would use the parser tag, while talking about how to use yacc or biosn or antlr to build a parser for the construction of an app. On the other hand, one would use parsing if trying to attack how to parse a csv file or other data formats.
Do compare the questions that show up in parser and parsing to see if the separation of ideas exists.
Thats just my take on it though. I could very well be wrong.
To create a synomym for a tag, navigate to the tag ( parser or parsing ) and click on the appropriate synonym link (that one is for parser).
You may also find https://softwareengineering.stackexchange.com/tags/synonyms to be of interest.