For the simple syrup: Combine strawberries and water in a large pot. Bring to a boil over medium heat. Reduce heat and simmer for 20 minutes. Remove from heat and strain berries with mesh strainer. Discard berry solids. Add sugar to remaining liquid and return to boil over medium heat; stirring to dissolve sugar. Reduce heat and simmer for 7-10 minutes. Remove from heat and let cool completely.
For the cocktail: In a glass with ice, combine prosecco, strawberry simple syrup, Aperol and club soda. Stir gently. Serve with lime wedges.