Yes — strawberries are vitamin-rich and loved by almost every parrot. Just wash them very well.
What it means for conures
There's nothing specific to a conure that changes the verdict on strawberries — it's treated the same as for parrots generally. A conure can handle a moderate portion.
The full picture on strawberries
Strawberries offer vitamin C, manganese, and folate, and their seeds and leafy tops are harmless. The only real concern is agricultural: strawberries consistently rank among the most pesticide-laden crops, so washing matters more here than for most fruit.
Wash thoroughly or buy organic; the tops are safe to leave on.
Halve or quarter for small species.
A messy fruit — schedule it before cage-cleaning day.