They can certainly apply, now I am a suspicious so and so and my thought was that they might be tempted to use it as a B, but I am not the person interviewing them.
That situation was resolvable, took time and money.
I look at it the other way around, if you have money you can afford to wait, if you do not you need to marry and file asap.
Do you have to go visa Utah, can you not just mary and file a CR1?
In a couple of years who knows what your plans will be and you can keep her GC alive for quite some time even if you want to carry on teaching.
Presumably your Parents can Joint Sponsor.
Unless he came up in the Lottery element you will not hear anything for sometime, a lot of people have applied and there is a long line.
Why did he have to leave his job to apply for HP?