"Normally" (in Europe?) the engine may turn on for heating purposes (even in models with an electric heater, as it can provide heat faster and more efficiently than the electric heater) but not for A/C purposes as the A/C totally depends on electricity and if there is sufficient SOC, why would the engine power up? Maybe, this behavior changes when you (auto)engage Save mode as Save mode aims at maintaining SOC. So, perhaps the engine is started to compensate for the loss of SOC caused by running A/C.
You said it was appropriate for the engine to start because it was cold. What I meant is, the trip you were going to make, was that outside or well inside EV range? Had it been well inside EV range, even taking into account SOC consumed by the heater, perhaps you would have decided it was inappropriate for the engine to be involved as it results in an unnecessary cold start and a waste of fossil fuels (let's not forget, if it is not too cold, the engine could actually be stopped before adding heat to the interior, because it could be caught up by the electric heater). Same when the engine starts for demisting, which if seems to do in your case.
I now realize the above is an odd question for you: for you, almost every trip is outside EV range, because you aim at maintaining SOC for as long as you can. You seldom run in to situations where you try to prevent the engine from running. Because you value your SOC more than most of us. Would it be fair to say: you bought a PHEV, when an HEV would have been more suitable for your needs and now you try to drive the PHEV as if it was an HEV?
If I had bought an HEV, for sure I would want the engine to start up for heating purposes as it would be stupid to use valuable SOC for that.
BTW: When the engine starts for heating purposes, it runs truly idle (low fuel consumption at low revs) as long as you are parked. It does not charge. When you start driving, the engine provides enough electricity for driving, but still it will not charge. Only when I hit the Charge button, it will start charging. I don't know what it will do when you engage Save mode. I expect the same, but when the engine runs for heating purposes, the electric heater (when present) will also be engaged. So, SOC will drop. And, like suggested before, the engine might kick in every now and then to recover lost SOC.
BTW2: When the engine starts for heating purposes (with sufficient SOC still available), the car will not engage parallel mode when you hit 40 MPH. Again, unless you hit Charge.