Yes, you really have to seal it with parafilm to prevent contamination especially when you incubate the plates, there is a very high chance of being contaminated without the parafilm.
I think it is not, I rarely do so and hardly see any contamination problem. I see that it depends your lab GLP and procedures you adopt while handling the plates.
There is no need to seal plate, at 37 oC plates are safe up to 72 hrs. (as per pharmaceuticals GLP) and petri plates are made as per Swans neck principle so, not to worry about contamination during incubation. Only chances of contamination is may be handling or during work.
I don't think sealing is required for the agar well assay, But then to prevent drying up of the antimicrobial agent in your wells, you can leave an open Petri dish with water or moist cotton to balance the loss of humidity in the incubator, in this way your solutions won't dry up. Generally the wells with antimicrobial agents don't dry, they only looks as such because the agent just diffuses into the agar.
In general, if you seal your Petri dish with Parafilm you will modify oxygen concentration and then the growth of your bacteria... de facto you have a risk of misinterpretation
But, clearly in your case I ignore the bacterial species you used and also the nature of your antibacterial agents