Combine water and rice in a medium saucepan. Bring to a boil. Reduce heat to low, cover and simmer until tender and most of the liquid has been absorbed, 40 to 50 minutes. Let stand, covered for 10 minutes. To cool, spread the cooked rice out on a large baking sheet and let stand until room temperature, then refrigerate until cold.
Heat a 14-inch flat-bottomed wok over high heat until a bead of water vaporizes within 1 to 2 seconds of contact. Swirl in oil, coating the bottom completely. Add beaten eggs and cook, tilting to cover the surface as thinly as possible to make an egg pancake. When the pancake is just set, 30 seconds to 1 minute, flip using a metal spatula and allow it to set for about 5 seconds. Transfer to a cutting board. Cut into bite-size pieces.
Swirl in half the oil into the wok, add garlic and shallots and stir-fry, using a metal spatula, until fragrant, 10 seconds. Add parsnips and Brussels sprouts, reduce the heat to medium-high and stir-fry until the vegetables are nearly cooked through, about 2 minutes. Swirl in the remaining oil, add rice, tomatoes and soy sauce and stir-fry, breaking up the rice, until heated through, 2 minutes. Sprinkle with cilantro, salt and pepper; add the egg pieces and toss to combine.