Calculate total impedance in RLC circuit with L and C parallel
According to my notes the total impedance is 1002 Ω, -86.6°


Your notes are correct and I think the mistake you may have made is adding 60 ohm (resistive) to the 1000 ohms (reactive) to get 1060 ohms. You need to add them as squares then take the square root hence \$\sqrt{1000^2 + 60^2}\$ = 1001.8 ohms.

The 1000 ohms is the reactive impedance of C in parallel with L. I'm presuming you know how to derive this because you appear to be making the right noises in your question i.e. the "(Xc*Xl)/(Xc+Xl)" bit is correct.