If partner told you that they want you to meet their parents does that mean they want marriage?

I don't really think so. Often it simply means that they live out loud and don't see the point of secrets and hidden meanings. The only reason my parents didn't meet the guys I dated was because they didn't want to. Too many kids, too many dates... they didn't want the drama. Other parents are eager to know everything.  One would suspect that by the time a person is called a "partner", they'd know the answer to this question. 

Not necessarily, but it does indicate 

a desire to get to know their partner 

a LOT better, and it certainly can 

lead eventually to marriage.

No.  But it does mean you are more than a side piece.

Yes, that is usually whats on their mind.

No, lol. It means they want to get you better. Or maybe it means they want to have cheaper dates, like Mom's home cooking. Don't try to rush things. Wait and see. You can't push the river; you have to just go with the flow. 

It often means that.

No.  It means they want a second opinion.   Or maybe they want to consider whether you are a possible life partner.  It is a step in a relationship, but not the last step.  Relax.