In a large saucepan, cook cauliflower florets in a small amount of boiling lightly salted water for 2 to 3 minutes or just until tender. Drain. Rinse with cold water; drain again. Set aside.
Remove fennel tops and stalks, save for another use. Slice fennel in half vertically to show the fennel heart; cut away and discard the solid core. Slice fennel into strips. In a large skillet, cook fennel in the 2 teaspoons hot oil about 3 minutes or until crisp-tender.
For dressing, in a small bowl, whisk together lemon juice and the 1/4 cup olive oil. Whisk in garlic, salt, and pepper.
In a large bowl, combine cauliflower, fennel, carrots, and onion. Add the dressing; toss gently to coat. Cover and chill for 2 to 24 hours. Just before serving, sprinkle with parsley.