I found this sentence interesting, because it helped me think about a distinction.
"Besétálok abba a parkba" certainly corresponds to "I'm walking into the park." "I'm walking in the park" is "Sétálok abban a parkban".
But the distinction with "fekszem abba az ágyba" versus "fekszem abban az ágyban" is somewhat different. You'd never "lie into" bed. I think I'd say "I lie in that bed" or "I'm lying in that bed" for both, depending a bit on context. I might be inclined to switch to future tense for the former "I'm going to lie in that bed" or even "I'm going to go lie in that bed".
I'm not entirely sure what the usage difference is, but guess it might be in the size of the park versus the size of the bed.
You could say, "I'm getting into bed," although that leaves out the word "lie." But that's what I usually hear when someone is talking about going from not-in-bed to in-bed.
I have started to put in into in all my answers in this exercise to get them right even though they are bad english.
You do not have to. Just keep it the way it should be, like me, to show that we are right, and we repeat and repeat and repeat the exercise.
Into the large park