Scanning the C language is done using regular expressions, typically. Parsing the resulting token stream cannot be done with only regular expressions.

At least not with the theoretical construct. Some "regular expressions" libraries, notably PCRE, let you do plenty that goes beyond the theory at the expense of some efficiency. Parsing some context-free grammars with these libraries is certainly possible, but you're not really restricting yourself to "just" regular expressions at that point. Regular expressions plus interwoven use of a stack is a pushdown-automata, which can totally parse C and HTML.

No, it can't. As in, you cannot parse HTML or C with regular expressions, and you can prove that that's the case. Regexes simply aren't powerful enough. This is one of the advantages of studying formal languages -- you get to learn about really useful models of computation (here, DFA's), but you also get to learn about their limits.