A white buildup has formed on stainless steel parts after soaking in NaOH with aluminium pieces How to remove?

Whatever the white stuff is, I would try some other common solvents first, acetone, alcohol, and see if they dont get it off. If not you could try a stronger acid mixture instead of just HCl.

Aluminium metal reacts with NaOH to form a number of substances. The most common is sodium aluminate, but it is also common to form the hydroxide - Al(OH)3. This latter product is insoluble in water, but it should dissolve in acid. The choice of acid is a problem. If you are using type 304 stainless steel - or whatever the local equivalent is - do not use HCl because this grade of steel is atttacked by chloride ions. If you are using type 316 stainless - you do not have a problem in this regard. I suggest that you use H2SO4 to be safe. You need quite a concentrated solution of acid - so something like battery acid should be oK. Give it time to react.
I do not know what you were doing, but mixing NaOH with Al is not advisable, as you have discovered. Also - unless you have specific training - be careful when dealing with concentrated acids. They are dangerous.