Wash the strawberries and carve out the stem. Place the strawberries in a bowl, sprinkle with balsamic vinegar and sugar, then toss gently.Preheat the oven to 350 F.
Cover a silicone baking mat with a baking sheet, wider and longer than the baking sheet, so no juice can escape. Lay the strawberries in the middle.
Bake for 30 minutes.
Remove from the oven, let cool slightly, and then transfer to a storage container until ready to use.
Strawberries will release some juice while baking, so when transferring them, make sure you pour all the juice.
In a glass jar, stir water and sugar until the latter is completely dissolved (warming up the water makes the process easier).
Cover with a lid and refrigerate until ready to use. This will make a bit more than necessary for the parfait.
Put a steel bowl in the refrigerator.
Pour the egg yolks into another steel (or other non-reactive material) deep bowl, and whisk them until light and paler. Warm up the sugar syrup to body temperature, then whisk it, one tablespoon at a time, into the egg yolks. Do not rush or the eggs will scramble.
Put the bowl over a small saucepan of barely simmering water (the bottom of the bowl must not touch the water). Heat the mixture, stirring every now and then with a wooden spoon to make sure that it does not overcook, both on the bottom and along the sides of the bowl.
When the mixture reaches 185 F and has thickened enough to coat the back of a spoon, take the bowl off the pan. This will take 20 minutes or so. Do not rush.
Using an electric hand mixer, beat at high speed for one minute, then at medium speed for 3-4 minutes and finally at low speed for 5 minutes. In the end, the beaten mixture will be thick: if you trail a ribbon of it over the surface, it will hold.
Place in the refrigerator for at least one hour.
When ready, take out of the refrigerator the empty bowl already prepared and pour the whipping cream in it. Beat until soft peaks form.
Take out of the refrigerator the bowl with the egg yolk and syrup mixture and gently fold the whipped cream into it, in four batches.
After folding the second batch of whipped cream, add half the amount of strawberry juice and fold it in, then repeat with the other half. Continue until until evenly mixed. The ending result must be homogeneous and airy.
Pour into a plastic freezer box, cover the surface with a piece of wax paper, place a lid on the box and freeze for at least two hours.
Take out of the freezer ahead of serving the parfait. If frozen solid, it will need about 15-20 minutes in the refrigerator to soften enough to be served.