My friends invite me to the party VS My friends invite us to the birthday party?

[Raymott]

No, Polyester.
If I said to you, "Would you like to come to a party?", you would not say back to me, "My friends invite me to a party."
This is my point. Your examples often, perhaps usually, have no practical application, as they are written.
tedmc has corrected the grammar, but it's still not a natural dialogue.
You might think that, but couldn't say it, because it wouldn't make sense. The following is normal:

Me: "Would you like to come to a party?"
You: [Thinking] "My friend is inviting me to a party. Will I go? Hmm, yes, I will."
You: [Speaking] Yes, I'll come. Thank you.

We would still not even think, "My friend invites me to a party" in English, because it's the wrong tense, even though it's happening in the present.

 

[Polyester]

I thought these sentences for two days.

The result is make no sense. Ok. I accept it.:up:

 

[Raymott]

The result is make no sense. Ok. I accept it.:up:

The result will never make sense until you study the usage of the present tense, present continuous tense, and present perfect tense in English.
You will find this in any grammar book, as well as all over the web.

Here are some pages to get you going, assuming you don't have a comprehensive grammar book:
http://www.grammar.cl/Present/Simple.htm
http://www.englishpage.com/verbpage/presentperfect.html
http://esl.fis.edu/grammar/rules/pressimp.htm
http://www.myenglishpages.com/site_php_files/grammar-lesson-simple-present.php#.VRDh5OFWrfc
etc.

 

[Matthew Wai]

If I said to you, "Would you like to come to a party?", you would not say back to me, "My friends invite me to a party."

If the speaker said it to a third person, it should be 'My friends (have) invited me to a party', because the action of invitation has already completed/was already in the past. Is it correct?
Not a teacher.

You: [Thinking] "My friend is inviting me to a party. Will I go? Hmm, yes, I will."

Both the OP and I would not think in English.

 

[Raymott]

Both the OP and I would not think in English.

That's beside the point. You can think it, but it's inappropariate in that context to say it.
You might not think in English but you, at least, Matthew, could write a sentence in which someone thought that in English.