Well partially it seems that if they're just parroting stuff back, they haven't really learned anything, so it's not much of a loss if you change your methods and they still don't learn. Basically, you haven't got a lot to lose. Mind you, I'm an engineer, not a teacher, and this is just my impression. I would defer to the actual experts.
I'd say just burn them over and over for parroting, and reward them over and over for thinking. If you ask questions of the class as the normal thing, a student will feel good about themselves for saying what you want to hear in response, which trains the teacher's password reflex. If you instead throw them curve balls all the time such that they reward themselves for solving the tricky puzzle or feel slightly ashamed for blurting out a phrase which doesn't connect to reality, they will learn to think. Maybe.
For stuff like multiplication tables, you really do want to just memorize verbal patterns a lot of the time. So obviously keep that in mind.
See my from first principles essay for an example of a teacher who I think did it right.
Basically, make the exercises require application of what you are actually trying to teach, rather than just a verbal statement of it.
For some introductory material, memorization really is required. 5 x 5 = 25 is a true fact that I expect a young student to simply memorize.
Looking back, we can understand why it is important to learn those types of things - but the explanation of why is often far more complex than the object level fact.
As a teacher, I wonder if it is possible to instill this skill into students the skills of rationality and critical thinking. I teach the third grade, and it is not immediately apparent how to apply this with my own class.
The problems I foresee are as follows:
In the sequences, it is suggested teachers should drill into students words don't count, only anticipation-controllers. How practical is this for an elementary school level? Also appreciated would be any ideas or experiences on how to do this, or how to combat the above problems. Hearing from other teachers would be excellent especially.