1. Population consisted of Jews and non-Jews of various religions and races. In Israel there are European Jews, Middle Eastern Jews, Sephardic Jews, Ethiopian Jews, Circassians, Arabs, Druzes, Armenians, Russians, and others. Similarly, in Khazaria the component elements were Byzantine Jews, Middle Eastern Jews, Circassians, Arabs, East Slavs, Kalizes, Ghuzz Turks, and others.
2. Jews in power. Khazaria's governors, army generals, and kings were mostly Jews. Israel's prime ministers, legislators, army generals, and mayors are mostly Jews. In both countries, the leaders possessed Hebrew names (in Khazaria names like Yitzhak and Yosef, in Israel names like Itzhak and Ehud).
3. Israel has an organized court system, just like Khazaria had.
4. Arabs attemped conquest of land repeatedly. In Khazaria Arab armies threatened Khazaria during the 7th and 8th centuries and almost took over the country and even briefly forced the Khazars to adopt Islam. In modern Israel many Arabs want to take over parts of Jerusalem and elsewhere.
5. Both Israel and Khazaria are at crossroads of civilizations, where trade-routes and peoples converge.
6. Both Israel and Khazaria celebrated Passover, Hanukkah, and the Sabbath.
7. Both Israel and Khazaria have been allied with Turks.
8. Both Israel and Khazaria are regarded as beacons of tolerance and freedom in the middle of other nations that don't share those characteristics.
9. Both Israel and Khazaria have had strong armies that have won decisive victories.
10. Both Israel and Khazaria have rich archaeological materials that have been explored at multiple sites.
11. Many parts of Israel and Khazaria are dry semi-desert or desert, but there are also agricultural areas.
12. Camels live in Israel and Khazaria.
13. Israel and Khazaria both border two major seas (Mediterranean+Red and Caspian+Black respectively).
I can also see other parallels!