All overtone modes are run in series resonant mode. Fundamental oscillation mode can be designed for series or parallel mode, most often parallel mode since no coils are required.
That's not quite right. Any adjustable oscillator will vary the load, so even with a series resonant crystal, the adjustment will take the load slightly to the inductive or capacitive side of the series resonant point.
Also, the company I ran made several thousand overtone oscillators that ran the crystals at about 10 - 12 pF load.
The oscillators were Colpitts, with an inductor and capacitor in the emitter circuit to prevent fundamental oscillation. Colpitts oscillators need a capacitive load in the emitter circuit, and the inductor and capacitor were tuned to somewhere between fundamental and third overtone. That meant that the tuned circuit appeared inductive at fundamental frequencies, so the oscillator wouldn't run like that. At third overtone, the tuned circuit appeared capacitive, allowing the circuit to run.
Like all Colpitts oscillators, the circuit from the base to ground has to be inductive. In my circuit, that was the crystal, running at about a 10 - 12 pF load.
I know that a lot of overtone designs run at series, and the inductors that are used to prevent the oscillator running at fundamental frequencies also make the crystal run at the series resonant point. Also, overtone crystals have much less pulling than fundamental crystals, so getting the load capacitance wrong with an overtone crystal will often only result in a small frequency shift, so it may go unnoticed.
20 pF load generally run about 20 to 50 ppm below series resonance
. A crystal will always run at a higher frequency when it is running with a load capacitance than when it is running at series resonance. However, the crystals are adjusted for a specific load capacitance, so a crystal adjusted for 20 pF will run lower than one adjusted for the same frequency but at series resonance, when running in the same circuit.
The frequency shift can often be a lot more than 50 ppm. We used to regularly make voltage controlled crystal oscillators that moved by +/-250 ppm minimum, so they changed by over 500 ppm.
Series mode operation does not imply the crystal is operating at its series resonant freq. Almost always it is operated below crystal series resonance to avoid crystal spurs that exist above crystal's series resonant frequency.
Series mode isn't actually distinct from parallel mode. Series mode is simply the point where the crystal's current and voltage are in phase. At slightly lower frequencies the crystal appears capacitive, so needs and inductive load, and at slightly higher frequencies the crystal appears inductive and needs a capacitive load. Voltage controlled oscillators will often change the crystal load from inductive all the way though series resonant to capacitive to get the pulling range, and there is no change in oscillation mode at series resonance.
Fundamental mode and all the overtones are distinct modes and only a few specialised oscillators can get more than one overtone running at any one time.
The spurious responses that all crystals have should always be above the correct crystal frequency, but running slightly below series won't help to avoid them because the spurious responses also have their series resonant point, and their frequencies also change with load, just less than the main response. For AT cut crystals, the design of the crystal rather than the oscillator is what stops the oscillator running at a spurious response. The spurious responses are not at predictable frequencies, but they should always have less activity that the main response.
SC cut crystals have a B mode just above the main mode, and the B mode oscillates more easily, but with terrible temperature coefficient. They need carefully designed filters to stop the B mode oscillating.