Whisk together miso, mustard, Sriracha, sake, and 1/2 cup plus 1 tbsp. mirin in a small saucepan. Add ginger. Bring to a boil, then lower heat and simmer 6 to 8 minutes, or until sauce is just thick enough to coat a spoon. Strain through a fine-mesh strainer, pushing against solids. Let glaze cool to room temperature.
Bring remaining 1 1/2 tbsp. mirin and the vinegar to a boil in a very small saucepan. Add goji berries. Return to a boil and remove from heat. Let cool, uncovered.
Put rack in middle of oven, then preheat oven to 400°. Line a rimmed baking sheet with parchment paper.
Cut 1 parsnip lengthwise into 8 spears. Cut remaining parsnips into 3/4-in. chunks. Transfer spears and chunks to baking sheet, season with salt and white pepper, and toss with sesame oil. Spread in a single layer, with spears cut side down. Bake parsnips 20 minutes, turning 2 or 3 times.
Meanwhile, bring a medium saucepan of water to a boil. Cut tofu into 4 rectangular slabs, each about the size of a deck of cards, and put in a large, shallow bowl. Pour hot water over tofu and let soak 15 minutes. Drain on a clean dish towel or a double layer of paper towels set on a plate.
Remove parsnip spears from oven when tips begin to brown; remove chunks when just fork-tender. Let chunks cool 5 minutes, then purée in a food processor with 2 tbsp. miso-mustard glaze and 1 cup hot water until smooth but able to hold a peak when spooned onto a plate (add up to 1/2 cup hot water if needed). Keep warm, covered.
Put tatsoi in a large saucepan with 1/4 cup hot water and cook over low heat, covered, stirring occasionally, until wilted, 5 minutes; drain and return to pan. Stir in goji berries and some of their pickling liquid.
Pour remaining miso-mustard glaze into a medium frying pan. Blot moisture from tofu with paper towels and add tofu to glaze. Bring to a simmer over medium heat and cook, spooning glaze over tofu, until glaze sticks to top of tofu and has reduced by one-half to two-thirds.
Spoon parsnip purée onto plates. With a spatula, set tofu on parsnip purée. Drizzle glaze over tofu and top with tatsoi and goji berries, then add parsnip spears.
*Find at Whole Foods Market, health food stores, and Asian markets.