Thermocouple noise causing spurious reading with SSR
Earth Grounding the Laptop thru VGA or the case can assist in diverting charger noise creating a large common mode noise to shift the PID control signal crosstalk to SSR and to thermocouple signal.  Perhaps the PID compensation needs to be adjust to make the system more stable and less sensitive to HF noise ( D gain )


You have not shown all the earth ground connections in your wiring schematic so it is not clear to you  why the ground loop is causing signal noise. 


The SSR , must be a ZCS type.  
You could add a CM line filter to the SSR output and consider shielding your uV thermocouple signal. 


The temperature control system exhibits a 20 second oscillation so it is clear the PID parameters are not optimized, but a secondary issue.

There are 100 or 120 half cycles per second available for switcher heater power, there ought to be plenty of time to anticipate the probe temp and regulate the heater but the PID output is clearly noise during the heating phase of the 20 second cycle when temp is below average.   The lack of spurious signals after 550s clearly indicates the SMPS for the laptop charger is contributing to thermocouple noise which is inverted to PID output.