In a 1-quart saucepan, bring the raisins and rum to a simmer over low heat and cook, stirring occasionally, for 2 minutes; remove from the heat. Let the raisins cool to room temperature in the rum, about 30 minutes.
Meanwhile, position a rack in the center of the oven and heat the oven to 350°F. Butter the bottom and sides of a 9x13-inch baking pan, line the bottom with parchment, and then butter the parchment.
In a large bowl, whisk the flour, baking powder, baking soda, and salt.
In a 2-quart saucepan, melt the butter over medium-low heat; remove from the heat and let cool briefly. Add the brown sugar to the pan and mix with a silicone spatula until thoroughly combined. Add the eggs and vanilla and stir until combined.
Add the sugar mixture to the flour mixture, stirring with the spatula until just combined. Gently fold in the raisins and any rum from the pan.
Scrape the batter into the prepared pan, smooth the top, and tap the pan on the counter once or twice to break any air bubbles. Bake until the top is golden-brown and a wooden skewer or toothpick inserted in the center comes out with just a few moist crumbs clinging to it, about 25 minutes. Let cool completely in the pan on a wire rack
Run a knife around the edge of the pan to loosen the blondies, invert onto a cutting board, remove the pan and parchment, and flip right side up. Cut into 16 pieces and serve.
You can store the blondies, covered, at room temperature for up to 5 days.