Wiktionary does not have its own API. MediaWiki, the software Wiktionary runs on, has an API, but is completely unaware of the structure and contents of the Wiktionary.
The best you can do is use the MediaWiki API to find the wiki page for the word you want, and then look at the table of contents. If the table of contents has a language section for the language you need and there is a Pronunciation section inside it, then use another call API to get the wikitext this section, which you will have to analyze yourself. It can use or not different patterns for different words, since Wiktionary is constantly evolving.
There are also Wiktionary mailing lists for the MediaWiki API .
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