Do Schengen visa officers check employment status for all applicants?

• You are young enough that being "between stages of tertiary education" and thus technically unemployed is quite common. It would be better if you were already accepted, but it is not necessary.
• Going to a professional conference is a good thing, especially if it ties into your studies. Previous Schengen visits are also good. You could have overstayed then and didn't do it, that makes you appear a lower risk of overstaying.
• A letter from your parents that you will return is completely worthless. They cannot make you return, even if they tried. If your parents provide the funding for your trip, you have to explain why they do it, and that the money is really a gift to you. This is a "source of funds" matter, not an "intention to return" matter.

Tell the truth, make the application, make it clear that you have a well-planned itinerary and that you can afford the trip. There is a risk that the application will be denied, but this is much less serious than getting caught in a lie.

Do the Schengen visa officers check the employment status for everyone?

You can never answer this question. It totally depends on the officer but the important thing to note is that it's irrelevant. If they do indeed check and find out that you are lying then it's considered as deception and this could mean that you will not be granted a non-immigrant visa or an immigrant visa in the future.

But saying that I had this thought of resigning exactly before going to Paris sounds fishy I think.

This could sound fishy but I don't think you have an option of concealing this information.

What should I do? My bank account has salary from my employer about a week back so just to reiterate, the documents seem perfect if I state that I'm still employed.

I would suggest you write a cover letter explaining your circumstances (your employment status) and also the conference invitation would put some weight on your application. If you have a good travel history and stable bank statements then I don't think you should be bothered about your current employment status. Just make sure you provide sufficient evidence that you have good ties to your home country and you have every reason to go back to India after attending the conference.

I'm from India and have often received Schengen visas. In your case, I believe its best to say you are unemployed in the form - in your covering letter, you can say (as a previous poster suggested) that you resigned from an earlier job. The fact that you have previous schengen visas and a valid US visa will greatly strengthen your application. Be sure to mention that in your covering letter.

Just say you are taking a break and apply. Don't lie on the visa application. Indian here as well. Once the visa officers even called up my company (TCS) to check if I was working there despite having multiple Schengen visits and UK visits. If you are from a smaller company, they'll certainly check.

I know it sucks with all the cancellation fees of flights, accommodation etc if a visa is rejected, but trust me lying is not worth it.

One of my visas was rejected by some racist officer from Malta. But since I didn't lie, I was able to both complain to the EU Commission as well. And the complaint helped me in the sense when I applied again to another EU country, shared the email chain, it went smoothly.