Molecular Orbital Theory and No. of bonds
Overall bond order equals number of electrons that occupy bonding orbitals minus numer of electrons that occupy antibonding orbitals divided by two.

Now you can apply the same concept indivudually:

For s molecular orbitals, both electron pairs "cancel out".

For p molecular orbitals, you have 2 electrons in sigma bonding orbitals and none in sigma antibinding orbitals, which means (2-0)/2 = 1 sigma bond.
Also, you have 4 electrons within pi bonding orbitals and only 2 electrons within pi antibonding orbitals, which means (4-2)/2 = 1 pi bond.
This yields the result that you expected.