pulsed to pulsed and steady output.

Like I said, your diagram is correct. Lurch's wiring isn't - the lamp will be on when the flashers flash off (if the parkers are on). Other than the out-of-phase wiring for bulbs, the only way to do it is a timed circuit that cuts bulb power from other sources - unless you want to rewire the flasher switch to energise DP relays (ie, before the flasher can); the relays connect the flasher can to the respective bulbs while disconnecting their other sources. (The basic circuit requires 1xSPDT and 1xSPST relay else 1xDPDT relay per side.)