Dodge, parry, sidestep, duck, shirk, fence, malinger are comparable when meaning to avoid or evade by some maneuver or shift.
Dodge implies quickness of movement or a sudden evasive shift of position (as in avoiding a blow or pursuit). It may imply artfulness, or craft, or clever deceit in evading not only the attack of an enemy but similarly the thrusts of a debater, or an examiner, or the demands of an authoritative power.
Parry does not imply dodging so much as warding off a blow and turning aside the weapon; it suggests skill and adroitness in defending oneself not only from blows with a weapon but from whatever threatens or proves awkward.
Sidestep comes very close to dodge in its suggestion of a quick maneuver to evade a blow or the facing of an issue; it usually suggests dexterous action in avoiding something imminent but often, on the other hand, suggests not ultimate avoidance but temporary delay or postponement.
Duck implies evasion or avoidance by or as if by a sudden bending or stooping.
Shirk implies evasion or avoidance by means that suggest meanness, cowardice, laziness, or sneakiness.
Fence suggests a maneuver comparable to one used in fencing; it may connote parrying, or thrusting, or guarding, but it typically implies more dexterity or more boldness in baffling inquiry than skill in warding off what is awkward.
Malinger implies, usually, a shirking (as of one’s duties or a hard task) through feigning illness or weakness.