Some of these have weird appended effects that just make the wording seem awkward. That's the only general critique I have, I think as concepts they're pretty interesting and I enjoy the theme, but they could maybe use slight adjustments for smoother phrasing, which I'll suggest below. Bolded suggestions mean the mechanics or numbers changed while italics are reworded but operate the same; reordered descriptions where the placement of sentences changes without any effect will be unmarked.
Falchion (Marth): Effective against dragon foes. Grants Atk/Spd+3. Restores 10 HP at start of turn and after combat.
Grants Atk/Spd/Def/Res+5 during combat and "Restores 10 HP after combat" to allies within 2 spaces. If unit is within 2 spaces of an ally, grants Atk/Spd/Def/Res+5 to unit during combat.
Slight nerf to Falchion's base effect so the two healing effects can be condensed into one sentence, whereas the Special Refine effect is just reworded to be more consistent with Joint Drive effects.
Aura: At start of turn, restores 10 HP and grants Atk/Spd+6 to unit and allies within 2 spaces for 1 turn.
If unit is within 2 spaces of an ally, grants Atk/Spd+6 to unit during combat, and, if ally uses magic or staff, reduces damage from foe's first Attack by 75%.
To make Aura's base effect wording smoother it's buffed to give Linde the Atk/Spd bonus as well. Instead of repeating "within 2 spaces," the refine effect now asks if the previously determined allies within 2 spaces are magic/staff units.
Hauteclere: Accelerates Special trigger (cooldown count-1). Neutralizes "effective against flying" bonuses. Deals +7 damage when Special triggers.
Restores 10 HP at start of turn. At start of combat, if unit's HP ≥ 25% *grants Atk/Spd/Def/Res+5 to unit during combat and neutralizes effects that prevent unit's follow-up attacks during combat. *
Just cleaned up the wording on the special refine effect and reordered some things for consistency. Note that you don't have to add an also for a second effect, only for a second condition.
Parthia: Grants Res+5. Effective against flying does. Restores 10 HP at start of turn.