You're right that present perfect is also OK, and often is the correct choice. Many students overuse the present perfect simple which is not that common in English. In this case, "started" has more the function of an adjective which describes the party. This meaning of "start" is not transitive and so it can't take a passive form. Most adjectives which are also past participles can only be used predicatively, not attributively ("a started party" is wrong).

Т.е. если already, то однозначно нужен perfect? Потому что действие завершено к некоему момент? И has и had подходит сюда?

Thank you for your answers. I found it on Facebook and saw couple times somewhere on the Internet. I guess they can't all be wrong.

It isn't correct. If you say 'The party's already started', the 's' is a contraction of 'has', not 'is'. 'Started' is a past participle, not an adjective.

Where did you see this written? I agree with you. I think it's either written incorrectly, or was originally written as "the party's already started" and then incorrectly substituting "is" rather than "has" for the contraction.