Regular expressions are strings formatted using a special pattern notation that allow you to describe and parse text.
"Parse" is a questionable word choice. A proper Regular Expression can only describe patterns found in "regular languages", though modern backtracking regex engines have a bit more power. Parsing implies a parser, which means matching a grammar somewhat more sophisticated than a regex engine can handle.
Regular languages can be described by regular grammars, and regular expressions are a notation for regular grammars. So you can indeed parse text with reference to a regular expression.
Most people would call a parser which only used strictly regular expressions a lexical analyser, lexer or tokenizer, but this is only a convention to simplify language definition and parser implementation when using simple techniques like LALR or LL.
But other uses may not need that level of sophistication. For example, a packet analyser in a router might use a dynamically updated state machine (a DFA), built from multiple regular expressions, for efficient dispatching of packets based on rules. I don't think it would be wrong to describe that component of the router as a packet parser.
For example, this (to me) blindingly obvious patent seems to be using a modified DFA to make routing decisions, and calls it a parser:
There seems to be this emerging pattern in my career where I stumble across all the people in the world who have decided to try and validate HTML using regular expressions.
http://stackoverflow.com/questions/1732348/regex-match-open-...
That poor bastard.
Really? So, we can parse HTML with regular expressions nowadays?
I think a more appropriate word choice would be "recognize" instead of "parse", with regard to the OP of the thread.