Yes, but as I stated, "Let us go to the park," also makes sense.

Not many people say it that way, i guess it's just the language evolving
every language has a unique way of saying it

spanish for example says vamos al parque (we go to the park) means the same thing

I understand that much, the thing I don't get is, if "Let us go to the park," makes sense, why can't "Mom, can you let's go to the park?"

its simple sentence structure
let us is a question (future tense) so the following sentence works
"mum can you let us go to the park" this implies you are asking your mum if you can go the the park
lets is in present tense so if you follow grammatical rules the sentence should be
"mum lets us go to the park" this implies that should you wish to your mum will let you go to the park it is a statement
another sentence could be
"mum lets go to the park" this is again a question this time it is asking your mother to go the said park
"lets go" is not the same as "let us go" so cannot be substituted
for example
"paul lets go of the rope" if you change lets for let us the sentence no longer makes sense
"paul let us go the rope" you cannot interchange the two as they mean different things (ever so slightly different but still)

also it is "lets" not "let's"
this is because "let's" means somthing that belongs to let
" lets build" a question asking to build
"let's build" a build belonging to let

But the way I was always taught in school, it was "let's." kjfhgkjhkghkdf

then whoever taught you shouldn't be a teacher

Maybe.
Like I said, I might just be being stupid right now, but it just doesn't make sense to me. /:

You know what else is puzzling?

when someone says "can't you do this?" "cannot you do this?" doesn't. make. sense.

Wut?
Isn't it supposed to be "Can you do this?"

Hm, weird. I've never heard it used like that.
Maybe because where I live the English language is fucked up anyway. xD

American?