The Mitzvah to build the Beis HaMikdash is one of the 613. The Rambam in the introduction to Sefer HaMitzvos emphasizes that the 613 Mitzvos are for all generations.
We actually don’t need the Parah Adumah to build the Beis HaMikdash — the last Gemara in Eiruvin says that it can be built by timmaim if there is no one tahor.
To bring Karbanos we don’t even need the Beis HaMikdash, just the Mizbayach.