It is not permissible for a man to only have beard in his chin as it implies "shaving" which is considered haram. As far as the trimming, the Muslim person should be considered as "having beard" according to common view.
Keeping the beard for Muslim men is considered both a prophetic sunna and a wajib act. According to a widely-reported hadith the Prophet said: "Let the beard grow and trim the mustaches". Other traditions bring similar meanings too. Shaving the beard with no valid reasons, like medical necessity or coercion, is therefore considered haram. Some mujtahids have said that beard should not be shaved as "ihtiyat wujubi" (obligatory precaution) which substantially does not much differ from a clear prohibition.
The growth of the beard that you described (covering the chin) is sufficient. Having the full beard is not wajib. Also, please note that on Shar'i issues, one has to fulfill what is incumbent even if the parents object to it.
Books on Islam, Muslims, Prophet Muhammad(s), Ahlul Bayt.
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