Cook quinoa: In a pot, combine the quinoa and water and bring to a boil. Reduce heat to medium-high and continue cooking until all the water has been absorbed, ~10-15 minutes. Remove from heat and let steam for another 5 minutes. Fluff with a fork.
Parsley Vinaigrette
Combine vinegar, mustard, salt, parsley, and brown sugar in a food processor. Stream in oil until emulsified.
Mix the parsley vinaigrette with the quinoa. Just before serving, toss quinoa mixture with greens, strawberries, and feta.