Not according to the games director. In every interview he was in about the new DMC, he states that the stories in the original DMC games were not important or a focus at all.
Even if he didn't care, it was still a "serious" story. You wouldn't call Devil May Cry satire or anything like that. Any statements to that sense are retroactive to the horrid voice acting.
Well ya. The story itself may be serious, but the characters were not. The story involves destroying/ruling the world. However the characters portray themselves in the face of it dictates the tone. They made Dante cocky and nonchalant, so the tone wasn't serious for the most part by design. Sure the serious moments weren't serious because the acting was terrible, but there's only like..... 2 serious moments in the game.
Not the point, I know that Trish and Lady were there but it was Kirie or whatever that held the spotlight rather than them and she was the one who I felt was way out of place.