In your code above, the two parfeval lines are initiating asynchronous function evaluation requests in parallel. You're also calling Function_I and Function_II directly in the client session on the subsequent lines, and that's what's being printed out in the command window. I suggest removing those two lines (i.e. function1=Function_I();), and instead add the following after the parfeval lines:
wait(f1); wait(f2); % Wait for parallel execution to complete
f1.Diary % Display the command-window output from the execution of Function_I
f2.Diary % ... and Function_II
You can also get the results using
result_1 = fetchOutputs(f1);
result_2 = fetchOutputs(f2);