We've never been to DLR, I'm hopeful that there is stroller parking similar to WDW. We have never been to WDW without a stroller...started going when girls were 2, 4, 6 (took a single stroller and double for the 4 and 6) -- fast forward they are now 8,10, 12 --> they don't need a stroller obviously, but I will be taking a nicer umbrella type stroller for my DS4, he can walk pretty far and most often has chosen to walk with us, but he's still young, he may get tired and we have 3 days total in the parks, we won't be going back for afternoon breaks (we never did at WDW either, for any trip). My kids have napped in their strollers long enough to recharge and everyone else keeps going.
Kills me how judgmental some people can be, in 2014 my then DD8 rode in the double stroller with my DS2, she's small for her age, but was just starting to get out of a knee brace for an injured patellar tendon (she's a gymnast). So she was in the stroller more than DD6 who occasionally took a seat, but she couldn't walk the miles you put on at a WDW trip with a knee that had been immobilized for several weeks prior. No one would know that at the time, because the brace was very hot and she didn't wear it unless she started to ache. Bottom line is, you never really know someone else's situation.
AND to be totally honest here, I don't look forward to when we're out of the stroller completely because that underside of the stroller is an awesome place to keep our water bottles, purchases, etc!!
Like mom2rtk says - 'do what's best for your family' - your fun depends on it!
Also, something I always do is take a small change purse sized "wallet" if you will...it holds my license, my disney visa, the gift cards for my children's spending and room key. That's it. And I make a somewhat "unfashionable" choice and wear cargo shorts/cargo capris. I can zip or button that wallet safely into a pocket and we're set, all the most crucial items are stored safely with me (my phone is in another pocket, but I find most rides at least at WDW, have a pocket to hold the phone if I don't feel secure holding it myself). For DLR I'm considering the lanyard for the tickets (I am a magic band lover ;-) )