It is a 6.2V zener, so anything over 6.2V and it will conduct. You have 1K from 17 volt so there is 11.3V drop across the resistor. Current drawn is 11.3/1000=11.3mA if there is no other component. Now if you put a 10K in parallel with the zener. it draw 0.62mA. BUT there are total of 11.3mA draw from the 1K, so 11.3-0.62=10.68mA still need to draw by the zener diode. So the zener still in reverse conduction and still hold 6.2V across. So the zener still regulate the voltage at the anode.
You can lower the 10K to say 1K. At 1K, it draw 6.2mA away from the junction of the anode. So the zener still conducting 11.3-6.2=5.1mA. So the zener diode still regulate the voltage.
But if the 10K is lower to 558Ω. At 6.2V, it draw 6.2/558=11.11mA. At this, the zener only draw 11.3-11.11=0.189mA. Almost to zero. The zener is at the edge of turning off. Any lower value than 558Ω, there will be no current into the zener and the zener turn off and the voltage will drop below 6.2V and you lost regulation.
This is called shunt regulation. The shunt regulator will draw the excess current that the load ( in your case the 10K) cannot draw. It will regulate the voltage until the load draw more than the current that the zener originally programmed to draw ( by the 1K). to increase the capability of regulation, you need to lower the 1K so at default, the zener draw more current. Then you will have more room for regulation.
For example if you reduce the 1K to 500Ω then the zener will draw 22.6mA to start. Then any load that draw less than 22.6mA will let the zener in regulation.