Ok here's how I see it.
God Slayer Magic on a basic level is superior to Dragon Slayer Magic.
Meaning if you use say the Dragon Roar vs the God Roar, God wins. However, apparantly through emotion, Dragon magic is apparantly capable of elevating. I imagine the same can happen with God Slayer Magic; however, this simply hasn't occured yet as the known God Slayers have been mostly cruel/Evil (Zancrow, Orga) and fought for their own pride and arrogance, and not for another. Chelia hasn't had the situation to defend Lyon or others to raise its level.
So technically, God is still > Dragon....no one has simply elevated it. Regardless, it doesn't effect fighting dragons.
Through plotness, Dragons somehow have a spell or curse or hex made on them to where they are immune to magic. However, Dragon Slayer Magic is not confined to this restriction, and therefore can harm dragons.
Sad really as I think that it should be more like Pokemon....regular magic should simply be "Not very effective, bit do damage" where Dragon Slayer Magic is normally effective or "Supper Effective".
That way people like Erza, Jellal, Jura, and Makarov don't look like complete fodder :/....at the same time though, I do like that it makes Dragon Slayer Magic important!