Familiares and parientes are almost synonymous, but there are some differences in usage: in Spain, familiares tends to refer to what the speaker considers their family members, as in the immediate/close family, i.e. brothers, sisters, parents, children and probably grandparents and cousins; parientes, meanwhile, would translate more as 'relations/relatives', including your extended family, i.e. distant cousins, aunts, uncles, etc. Allegados is also used to refer to anyone close to the person, including family and friends.
-Voy a visitar a algunos parientes/familiares a Barcelona — I'm going to visit some relatives in Barcelona
-¿Hay algún familiar a quien podamos llamar para decirle que estás aquí en el hospital? — Is there a family member/relative we can call to tell them you are here in hospital?
-Los Fernández son parientes míos por parte de mi madre — The Fernandez family are relations of mine, on my mum's side
-Es un pariente (familar) lejano — He's a distant relative
-Tengo que hablarlo con la parienta — I have to pass it by the missus/wife [Colloquial]
-Debido al brote de coronavirus, solo tres familiares o allegados pueden asistir al entierro — Due to the coronavirus outbreak, no more than three friends or relatives may attend the burial