In a 3- to 4-quart pan over medium-high heat, bring milk to a simmer. Stir in sugar, rice, butter, and salt. Reduce heat, cover, and simmer, stirring occasionally, until mixture has thickened slightly and rice is very tender to bite, about 45 minutes.
Meanwhile, in a small bowl, sprinkle gelatin over 2 tablespoons cold water; let stand until soft, about 10 minutes. Stir into hot rice mixture, followed by rum and vanilla. Chill uncovered until cool but not set, about 1 hour.
In a bowl, with a mixer on high speed, whip cream to soft peaks. Fold cream into rice mixture. Divide pudding evenly among six parfait glasses (at least 8 oz.). Chill until set, at least 1 hour, or cover and chill up to 1 day.