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 the whisk attachment, whip the heavy cream and vanilla on high speed until soft peaks form. Fold the whipped cream into the pumpkin mixture.
To assemble, spoon some of the pumpkin mixture in parfait glasses, add a layer of whipped cream, then some chopped cookies. Repeat, ending with a third layer of pumpkin mixture. Cover with plastic wrap and refrigerate for 4 hours or overnight. To serve, decorate with whipped cream and slivered crystallized ginger.
Sweetened Whipped Cream (Makes 2 Cups)
Pour the cream into the bowl of an electric mixer fitted with the whisk attachment. Add the sugar and vanilla and beat on high speed until the cream forms soft peaks.
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