Translation:The rich politicians are flying on new airplanes.
i would say 'on new planes' not 'by new planes'. you might say 'i travel by plane' but that's a more categorical statement. i travel "by foot" but i wouldn't travel "by new feet" even if i'd had them replaced lately....
my knowledge is limited but I thought the definite article was always used
The definite article is only used if you're talking about definite things. The airplanes in this sentence are just some new airplanes, not the new airplanes.
Your translation is strange. Better would be: The rich politicians fly in new airplanes.
"On new airplanes" is the correct and literal translation here, right.
"By new airplanes" sounds a bit off (as if they are passing the planes), "with new airplanes" is more natural, and these phrases have the literal translation "új repülőgépekkel".
Just a comment on the English translation. I find the use of "with" or "on" in English depends on which form of the present tense is used. With "they are flying" I would use "on", as this form of the present implies they are actually in the air (or will be soon). It's a one-off situation. If we mean "they fly", meaning generally, it implies the new airplanes are like a tool. I could use "on", but "with" would be more natural for me.