If you are asking about the Sunan ar-rawatib السنن الرواتب (the sunnah prayers that are related to fard prayers) like the fajr prayer, the sunnah before and after dhohr, before 'asr, after maghrib etc..
Then you can only start praying them after the beginning of the time for the related (fard) prayer, and of course in case they are sunnah ba'adiya سنة بعدية (sunnah after the fard) of course you must pray the fard prayer first, but it seems you've asked about the sunnah al-Qabliyah السنة القبلية.
Note that the above is about actual prayers, as some madhhabs allow praying qada' of sunnah prayers which would be an exception of what I've said earlier.
Usually the time of the prayer starts with the call for it (Adhan), with the exceptions of a possible first call for fajr in Ramadan and the Jumuah prayer which must not necessary be called for at the beginning of the time for dhohr.
See also What are Sunnah Rawaatib?