Is JSON.parse supported by all major browsers?
Pretty much, yes (source). Even IE8 has it (provided you're not in IE7 emulation mode). If you need to support IE7 and earlier, read on.
No, older browsers (IE7 for instance) mostly don't have it. (More: http://caniuse.com/#search=JSON.parse)
However, just a small script is all you need. The inventor of JSON, Douglas Crockford, has no fewer than three for you to choose from on his Github page:
json2.js: Provides both
JSON.stringify. Parsing uses a few regexes to defend against script injection attacks and then passes the result to
eval. This isn't generally considered a very good idea.
json_parse.js: A recursive-descent parser that doesn't use
json_parse_state.js: A state-machine parser that doesn't use
Use what suits you. :-)