Many characters are made up of a semantic and a phonetic part.
Is the phonetic part only valid with regards to Mandarin pronounciation?
Or not even for Mandarin in all cases?
Or does it work with for example Cantonese aswell? That is are chinese dialects at least partially "parallel"?
I suppose the phonetic clue does not help a Japanese or Korean learner. Am I right?
Generaly yes. But I want to mention that the Mandarin was also a dialect. So I'm not agree that the dialects are variations of Mandarin, I think that they are pararrel. Most dialects are more ancient and closer to the ancient Chinese than the Mandarin.
To answer your question: Yes, the phonetic part is valid in all the dialects since the dialects are just the variation of Mandarin Chinese, in which ancient Chinese people drew upon or gave ideographic meaning to various objects and concepts in the environment. So in other words, Chinese dialects are "parallel" in a way.
Most Chinese characters have been around a lot longer than modern Mandarin, so their phonetic parts often reflect much older pronunciations to the extent that these hints are actually not quite as reliable in modern Mandarin.
Take, for instance, 工, pronounced gōng in Mandarin; it is used as a phonetic in the characters 江、紅、項, but they are now pronounced jiāng, hóng, and xiàng, respectively, even though their pronunciations were much closer in older varieties of Chinese. Cantonese, for instance, is considered a more phonologically conservative variety of Chinese that preserved more of the rimes from Middle Chinese. As such, 工、江、紅、項 in Cantonese are pronounced more similarly: gūng, gōng, hùhng, hohng (using Yale romanization).
In general, all of the Chinese varieties each have underwent their own set of sound changes. Because sound changes are typically systematic, words that were pronounced similarly tended retain their similarity even as their underlying pronunciations changed so the phonetic parts of characters are often still relevant. However, as shown by the example, sometimes initial slight differences between them could also end up being magnified over time.
Regarding your question about Korean and Japanese, while they are not related to Chinese, they did initially borrow the pronunciations that came with those characters as well when they borrowed the vocabulary from Chinese. In the case of Japanese though, they assigned their own non-Chinese readings to many characters, so any phonetic hints in the original characters would no longer apply to those readings.
The Korean language (to my knowledge at least) used Chinese characters only for Sino-Korean words of Chinese origin, and did not assign non-Chinese readings to them; for this reason, the phonetic hints for Chinese character readings in Korean are still relevant. For instance, according to Wiktionary, the pronunciations of the four example characters above in Korean are gong, gang, hong/gong, and hang.
Note that an added complication is that different words may have been imported into Korean and Japanese during differing periods, so their pronunciations will reflect the state and variety of Chinese used at that time. This may also end up making the phonetic hints not as reliable.
