Your second interpretation is correct: you parry right after the opponent attacks. If the parry is successful, you reveive no damage and take another turn; if it's unsucessful, you take the damage and it's your opponent's turn.
You're also correct in that an attack of 1 would always be parried successfully. You could house rule it so that declaring "parry" would be enough, unless:
A. The attack is a critical hit, which cannot be parried.
B. You're charging, in which case you must receive the damage and then attack.
C. You have a defense of 1, and, for tactical reasons, you'd prefer to reroll your defense.
Keep in mind, however, that Krig: Intervention matches shouldn't be played with that house rule. If you achieve a perfect parry (rolling a 1 on your parry d4), you'll gain a +1 bonus on your next attack.
Also, practically speaking, a player may forget to parry an attack of 1, in which case their error results in damage taken.