They succeed in part by insulating their children from Western culture (no TV, no Internet, little interaction with outsiders), and by giving them very little in terms of secular education, so that getting a career is not really an option for most.
Unfortunately, for the ultra orthodox, yes. The parliamentary system is a majority coalition, and more often than not, the ultra orthodox parties are the kingmakers, so they get a lot more influence than one would expect.
Specifically, they are exempt from the standard Israeli STEM + foreign language curriculum (which is quite comprehensive - to graduate highschool, you need passing grades in standardized tests that includes -- among other things -- English, Math, at least one of biology/physics/chemistry, literature, history, Hebrew, sometimes bible studies -- depends on the government at the time). You have to have 20 credits among these - at least 3 in English, at least 3 in Math, and there are more requirements).
About 10 years ago, there was a government they were not part of, and this exemption was removed ; but the following cycle, they were kingmakers again and it got reinstated.
Edit: Added some details about the Israeli curriculum.
