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Place the rum in a heat-proof bowl and sprinkle the gelatin over it. Set aside for 10 minutes for the gelatin to soften.
In a large bowl, whisk together the pumpkin, granulated sugar, brown sugar, egg yolks, orange zest, cinnamon, nutmeg, and salt. Set the bowl of gelatin over a pan of simmering water and cook until the gelatin is clear. Immediately whisk the hot gelatin mixture into the pumpkin mixture. In the bowl of an electric mixer fitted with a whisk attachment, whip the heavy cream and vanilla until soft peaks form. Fold the whipped cream into the pumpkin mixture.
To assemble, spoon some of the pumpkin mixture into parfait glasses, add a layer of whipped cream, then some chopped cookies. Repeat, ending with a third layer of pumpkin. Cover with plastic wrap and refrigerate for 4 hours or overnight. To serve, decorate with whipped cream and slivered crystallized ginger.