Reason of wars between Romans and Sassanids?

 Pretty much what Zagros said, the Sassanids intended to reclaim the ancient frontiers of the Achaemenid empire, a letter from Shapur II to the Emperor Constantius II dated 359 states this intent;

 "Shapur, King of Kings, brother of the sun and moon, sends salutation... Your own authors are witness that the entire territory within the river Strymon and the borders of Macedon was once held by my forefathers; were I to require you to restore all this, it would not become ill of me...but because I delight in moderation I shall be content to receive Mesopotamia and Armenia, which was frauddulently extorted from my grandfather...I give yuo warning that if my  ambassador returns empty-handed I shall take to the field against you, with all my armies as soon as the winter is past"

 Obviously Constantius refused, but theres no doubting that if Shapur had the chance he would attempt to restore Persia to the heights of the Achaemenids 8 centuries earlier. If necessary he would use his armies to reclaim these territories from Rome. I doubt this was possible the Sassanids had never really threatened that far into Imperial territory before and even though Persia was in the midst of Golden age and Rome in decline, Rome still had strength enough to hold its own.

 The rivalry between these two empires was usually confined to Mesopotamia, Syria and the constant tug of war in Armenia and only in the 7th century as far as i'm aware did it seriously spread into Egypt and deep into Asia Minor.