Set a fine-mesh sieve lined with dampened cheesecloth or white paper towl over a bowl, then spoon the yogurt in & strain in the refrigerator for 8 hours or overnight.
When strained, transfer the yogure to a large bowl & stir in 1/3 cup of the sugar & the vanilla extract.
In a small bowl, sprinkle the gelatin over the water & set aside to soften.
In a small saucepan, warm 1/2 cup of the mango puree over medium heat, then stir in half of the gelatin mixture & continue stirring for about 30 seconds to dissolve the gelatin.
Stir in the remaining mango puree & the lemon juice, before removing from the heat.
Transfer the mango mixture to a bowl set in a larger bowl of ice & water & chill, stirring frequently, until thickened & the mixture starts to mound.
Remove the bowl of thickened mango puree but keep the bowl of ice & water ready for the raspberry mixture.
Meanwhile, in another small saucepan, warm 1/2 cup of the raspberry puree with the remaining 1/3 cup of sugar. Stir in the remaining gelatin mixture & continue stirring for about 30 seconds to dissolve the gelatin.
Stir in the remaining raspberry puree, then transfer this raspberry mixture to a bowl & quick-chill it over the bowl of ice & water as was done with the mango mixture.
Divide & layer the mixtures among six 5-ounce parfait glasses, 1st layering the raspberry mixture then the mango & finally the thickened yogurt.