Place watercress in a sieve, pour boiling water over to wilt it, and drain on paper towels. Combine wilted watercress with lemon zest in a small bowl. Mix cream cheese, garlic, dill, and parsley together in a separate bowl.
Roll out 2 puff pastry sheets gently on a work surface until slightly larger than length of salmon fillets; cut sheets in half. Place 1/2 watercress mixture in center of each sheet. Spread 1/2 cream cheese mixture over watercress. Center salmon fillets on top of cream cheese mixture; squeeze lemon juice over each.
Brush the edges of the pastry with some of the egg. Fold top and bottom of pastry over the salmon; trim edges to remove excess pastry. Press pastry seams together with your fingers to seal. Transfer wrapped fillets, seam-side down, to baking sheet. Brush top with beaten egg.
Bake in preheated oven until pastry is golden brown, 20 to 25 minutes.