One of the oldest of these is Nemroz or Nowroz (various pronunciations and transliterations) from the Persian world, celebrated by people of all religions or none from that ancient part of the world. But the Pascal celebrations (Passover and Easter) will most likely be the most common among us, again whether or not we practise the respective religions they relate to.
Of course Passover, commemorating a hasty escape and time in the desert, has very specific food requirements, especially no leavened bread or other leavening (I'll leave the specifics to others), while Easter, which grew out of Passover (Last Supper a Seder of course) has seen many "pagan" pre-monotheistic religious and cultural practices adapted to Christian rites in different parts of the world.
And for some of us, south of the Equator, we are carrying a Spring festival's customs into autumn, as many peoples of European origins did with winter dishes at Christmas and New Years, in the heat of a summer in Australia, South Africa, or the southern tip of South America!
Though this has been changing in recent decades...
I was trying to think of some more specifically "food" festivals and none comes to mind for the moment. That's probably because many festivals celebrating food are in mid to late summer ("harvest time").
Bixa, the Persian festival Nowruz (various spellings and pronunciations) is also an ancient Spring Equinox celebration, and has spread as far west as the Balkan peninsula. en.wikipedia.org/wiki/Nowruz
I mostly think of zeppole (choux pastry filled with sweetened ricotta) for san Giuseppe, but I see that in NOLA, it is quite the celebration!
No, of course the Pascal celebrations are not really food-centred, as they commemorate sacrifice after all, and Passover means not eating normal bread, the staff of life in the Mediterranean. Of course Easter became conflated with older fertility rites in Christianized countries. But in colder temperate zones there isn't really much seasonal food yet, except for the poor superfluous boy lambs (snif).
Bixa, that Spring Festival seems to be the same as what we call Chinese (Lunar) New Year, or Tét in Vietnamese.
The Easter chocolates arrived at the supermarket just this week, which is extraordinarily late. I suppose the marketing experts have learned over the years that people will not buy this stuff until the second half of March even when Easter is extraordinarily early.