(That's a good question! Lemme double-check the rules.)

(It's tempting to say the Parallel Falchion would need its Overdrive triggered to grant its character actions within Stopped Time. But the rules are clear! All cards that can create Stopped Time are also able to Act during Stopped Time.)
(I could still rule otherwise if I wanted to. After all, the Falchion's what creates the Stopped Time, not whichever character it gets attached to, so it wouldn't be unreasonable to say: "The character with the Parallel Falchion does not themselves create Stopped Time, and therefore cannot act within Stopped Time.")

(But that'd be boring. More to the point, that's not entirely how I'd rule Attachment cards work. Unless some other effect calls for them separately, I consider a Character with an Attachment to be one entity with the effects of both cards. So whoever gets the Falchion can act freely in Stopped Time.)

(Voting remains open for now. I won't lock in JUNE'S TURN until she has about a full turn's worth of ACTIONS lined up, unless one of those actions creates new information you'll want to react to.)