Ivan, if it is part of your departure clearance, you must read it back in full. If something along the lines of what you post is given to an aircraft in flight, you are correct, the latter part is not a clearance, so there is no requirement to read it back.
So, if you are given "ABC123 is cleared flightplan destination via XYZ, after departure climb straight ahead flight level 120, when instructed by radar climb flight level 340, frequency when advised 123.45, squawk 1234", you read ALL of it back.
FWIW, the phraseology you give is incorrect.
"
Climb flight level XXX, when instructed by ACC climb flight level YYY" is more correct - though I don't promise that's 100%!
"
Level" is a description of the aircraft attitude - i.e. not climbing or descending. Therefore in the above phrase, the words are "
flight level". Also the word "
to" in a climb or descent clearance is only used when the level-off point is an altitude or height and then the phraseology is "
Descend to altitude 2000 feet XXXXXX QNH/QFE (as appropriate)
1004" or similar.