Preheat the oven to 350 degrees F. Brush an 11-by-17-inch jelly roll pan or rimmed baking sheet with vegetable oil and line with parchment paper; brush the paper with oil and dust with flour, tapping out the excess.
Make the cake: Whisk the flour, baking powder, baking soda and salt in a medium bowl. Whisk the granulated sugar, vegetable oil, milk, eggs and vanilla in a large bowl. Add the flour mixture in 2 batches, whisking until incorporated. Transfer half of the batter (about 1 1/2 cups) to a separate bowl. Tint one batch red and the other batch blue with the food coloring.
Spread the red and blue batters across the width of the prepared pan in alternating stripes, making each stripe about 2 inches wide (don't worry if the stripes aren't perfectly straight). Bake until the top springs back when pressed, 12 to 15 minutes. Let cool about 3 minutes in the pan. Dust with confectioners' sugar and spread a clean kitchen towel (not terry cloth) on top. Place a cooling rack on the towel, then invert the cake onto the rack; peel off the parchment. Roll up the cake and towel together, starting at a short end. Let cool completely on the rack, about 1 1/2 hours.
Make the frosting: Beat the butter and salt in a large bowl with a mixer on medium speed until creamy, about 1 minute. Add the marshmallow cream and vanilla; beat on medium-high speed until fluffy, 2 to 3 more minutes. Beat in the confectioners' sugar, then beat in the milk (the frosting should be spreadable).
Gently unroll the cake (don't worry if the ends are still curled up). Spread with half of the frosting all the way to the edges. Reroll the cake (without the towel) and trim the ends. Cover with the remaining frosting.