Prepare the quinoa according to the package directions. Set aside to cool.
In a small skillet over medium-low heat, melt the butter with the garlic. Turn off the heat and allow it to sit for 5 minutes.
Arrange the carrots, parsnips, red onion and squash on a large rimmed baking sheet. Pour over half of the garlic butter, sprinkle on some salt and pepper and toss to coat. Roast the vegetables, tossing occasionally, until they're nice and deep golden brown, 35 to 40 minutes. Set aside to cool slightly.
Pour the other half of the garlic butter into a bowl (reserve the skillet for the pine nuts). Add the lemon juice and some salt and pepper and whisk.
Add the pine nuts to the reserved skillet, set over low heat and toast, tossing occasionally, until light golden brown, 5 to 7 minutes. Set aside.
To assemble the salad: Add the arugula to a large bowl, followed by the roasted vegetables, buttery dressing, quinoa, pine nuts and Parmesan shavings. Toss it all together and serve while still warm or at room temperature.