Your opponent cannot activate Quick-Play Spell Cards.
Rulings
Individual Card FAQs
"Invader of Darkness" prevents the activation of Quick-Play Spell Cards, but doesn't negate their effects. So if your opponent activates a Quick-Play Spell Card and you chain "Call of the Haunted" to Special Summon "Invader of Darkness", he does not negate the activation or effect of the Quick-Play Spell Card.
DARK|
Level 8|[ Fiend / Effect ]
ATK 2900|
DEF 2500