Translation:He runs every day in the park near his house.
"He runs at the park near his home every day." wrong, eh? More adventures in obvious alternate translations not being accepted.
"near" "by" and "nearby" should all be acceptable. Also, translations using "the" instead of "his" should also be accepted, the house isn't explicitly described as his.
According to the Oxford dictionary, “nearby” is not (commonly) used as a preposition.
Exactly. It's a very common Chinglish mistake to say things like "nearby his house". Speakers of some other languages learning English do it too.
On the other hand, "close to his house" should be accepted.