Let's start with a simple statement: Italian taxes are something you should avoid to mess with (on your own!) as long as you are alive.
Every year, by the end of February, SISSA will provide you with a so called "CUD", a declaration that states how much SISSA has paid you during the previous year and how much taxes the School has deducted for you in the same period. Hence all the income you receive from SISSA is already "taxed" as SISSA pays taxes for you. If your sole income comes from SISSA, you'll have to download the file and keep it as a reference or, if you are Italian, give it to your family in order to include the relevant infos in your parents' declaration. If this is not your only income, then you'll have to fill an income tax declaration. As this procedure is far more complicated than the quest for a renormalizable theory of quantum gravity, we strongly advise you to ask for help from one of the centri di assistenza fiscale in Trieste. These offices are usually run by workers' unions and have a special status that enable them to deal with your file with the Ministry.
For Italians only
You PhD fellowship is regarded to be an employee-like income, hence these rules apply.
The PhD fellowship is not subject to personal income tax deductions, according to art. 4 of Law n.476, dated 13.08.1984
At the moment a PhD student is regarded as a university student and hence our families are entitled to get a partial tax reimbursement for the rent expenses. (E.g. the answer given by this tax expert - look for affitto)