A DPDT switch and a DPDT relay to do what could be achieved with a SPDT switch is certainly the "Steamroller to crack a nut" school of design, but effective as a demonstration!
The circuit looks fine to me, assuming a standard pin-out octal relay:
Coil on #2 & #7.
First contact set:
Common #1, n/c #4, n/o #3
Second contact set:
Common #8, n/c #5, n/o #6.
With the DPDT switch in the up position (up with respect to the tags as drawn) there's no power to the relay coil, and you have a circuit completed to the series lamps via the n/c contacts on relay pins 1 & 4.
With the DPDT switch down, the relay is energized and the circuit is completed to the parallel lamps via n/o contacts on 6 & 8.