To a saucepan, add the blackberries, sugar, water, and thyme. Bring to a gentle boil. Once the blackberries have started to soften, gently mash them by a wooden spatula.
Adjust the heat to low and let the mixture simmer uncovered for about 20 minutes, stirring occasionally.
Strain the mixture into a bowl. Press the mixture to get as much liquid as possible. Let it cool.
In a glass, put ice cubes then pour sparkling water. Then pour the blackberry syrup and squeeze lemon juice into it. Give it a stir, garnishing with fresh thyme and a slice of lemon. Please adjust the amount of syrup and lemon juice to your liking.