Translation:There is a park near the house.
Why the "есть" here? What does it add or emphasize?
есть means is.
Yes, but we usually omit есть in a sentence like this.
Near the house is a park is not accepted.
Does it roughly translate to "in the vicinity/neighborhood of the house there is a park"?
Yes. That is an acceptable, literal translation. DL should accept it but often does not accept literal translations, for reasons I've never figured out.
Can "есть" be omitted here?
Yes. See comment above.