No, they don't have the same meaning.

Tonight, I'm having dinner with my parents.  - This is a commitment, and it will be very difficult to change your plans.

Tonight, I'm going to have dinner with my parents.  - This is simply a plan, and it's possible that you haven't even talked to your parents about it yet. The key difference is that you can easily change your plans.

Although native speakers freely use both, we certainly notice the nuance!

Agreed. They are different. If the arrangement is not fixed, we can use either form.

A: Got any plans for the weekend?

B: Well, I'm going to have dinner with my parents tonight, and then go camping on Saturday.

A: There's a really cool party on tonight.

B:  Is there? Hmm. Maybe I'll call my parents and tell them I'll have dinner with them Sunday night instead.

However, if it's fixed, we use the present continuous.

A:  "Do you want to come to a really cool party tonight?"

B: [disappointed]. "I can't. I'm having dinner with my parents tonight. Sorry."

This means that that dinner with the parents is definite. A native speaker would not use a 'going to' construction, because the speaker knows that he or she cannot get out of the dinner arrangement. If the arrangement cannot be changed, the present continuous is the only natural option.

This is not a question of time or simplicity. Remember that English is a stress-timed language, and it actually takes pretty well the same time to say 'I'm going to have dinner' and it does to say 'I'm having dinner'. In natural native-speaker speech, the first words of the former are merged into something like 'aimgnuhv', a sequence of sounds which is barely a millisecond longer than 'aimhavin'. Our choice of verb form depends on our attitude towards the certainty of the event, not the time or effort it takes to use a particular construction.

Stephanie's answer is excellent."I'm going to have..." does leave a little uncertainty, and the difference can be pleasingly subtle.

"Going to have dinner" gives the faintest chance that I will change my plans if you have a better offer! I'm having dinner is much more final.

For me, the more sure your plans are, and the closer they are, I would say stick to the present continuous.  Thus, I think #1 is better in the example you gave above. Of course, native speakers will constantly use the two interchangeably, but why bother with the extra words when the first one is so simple and understood.  What is simplest is best.  The second example I would say sounds better with a little more uncertainty:  I think I am going to have dinner with my parents tonight.  Do you see how that changes things? 

By the way, always remember to capitalize the personal pronoun "I"...

I hope I helped.