Is it forbidden for a girl to talk to her male cousin if the conversation is free of words that displease Allah, and is intended for maintaining kinship ties and guiding him to good, and is this forbidden without the parents' knowledge?

The default rule for a woman speaking with a non-mahram man is that it is not permissible unless there is a need, and it is stipulated that her speech be free from being overly soft and seductive, free from suspicion, and safe from temptation. This is in accordance with His, the Exalted's, saying: "So do not be soft in speech [to men], lest he in whose heart is disease should covet." It is advised not to converse with a non-mahram young man, whether for kinship or for da'wah, especially in the absence of mahrams, to avoid temptation, ill-conjecture, and situations of doubt.