Combine the golden beets, parsnips and sweet potatoes in a heatproof mixing bowl, and combine the red beets and onions in another heatproof mixing bowl to prevent any bleeding from the red beets.
Combine the vinegar, sugar, salt, mustard seeds, celery seed, peppercorns, garlic and 1 cup water in a saucepan. Bring the mixture to a boil over medium-high heat to dissolve the sugar and salt.
Pour some of the hot pickling liquid over the red beets and onions just until they are submerged, and then pour the remaining liquid over the golden beets, parsnips and potatoes. Let the vegetables sit in the pickling liquid until just tender, about 1 hour.
Drain the vegetables and plate the red beets and onions. Top with the other vegetables just before serving, and garnish with the goat cheese, parsley and pistachios and drizzle with olive oil.