While it seems like a reasonable question, in fact a native speaker would very rarely this:

He went running to the park

Why? Because there are alternatives which are much clearer:

1. He went running.
2. He went jogging, at the park.
3. He went to the park.
4. He went to the park, to have a run.
5. He went to the park, to go jogging.
6. He ran to the park.
7. He suddenly jumped up, and ran to the park.

It's difficult to imagine a case for "He went running to the park" but you are correct that it sounds ambiguous when considering those two meanings of "go". It could be either of the choices you mentioned, so it's better to avoid such a confusion.

P.S.

"go + Ving". Do you mean "go + ing"? That is, "going".

While it seems like a reasonable question, in fact a native speaker would very rarely say this:

He went running to the park

Why? Because there are alternatives which are much clearer:

1. He went running.
2. He went jogging, at the running track near the park.
3. He went to the park.
4. He went to the park, to have a run.
5. He went to the park, to go jogging.
6. He ran to the park.
7. He suddenly jumped up, and ran to the park.

It's difficult to imagine a case for "He went running to the park" but you are correct that it sounds ambiguous when considering those two meanings of "go". It could be either of the choices you mentioned, so it's better to avoid such a confusion.

P.S.

"go + Ving". Do you mean "go + ing"? That is, "going".

While it seems like a reasonable question, in fact a native speaker would very rarely this:

He went running to the park

Why? Because there are alternatives which are much clearer:

1. He went running.
2. He went running at the park.
3. He went jogging, at the park.
4. He went to the park.
5. He went to the park, to have a run.
6. He went to the park, to go jogging.
7. He ran to the park.
8. He suddenly jumped up, and ran to the park.

It's difficult to imagine a case for "He went running to the park" but you are correct that it sounds ambiguous when considering those two meanings of "go". It could be either of the choices you mentioned, so it's better to avoid such a confusion.

P.S.

"go + Ving". Do you mean "go + ing"? That is, "going".

While it seems like a reasonable question, in fact a native speaker would very rarely this:

He went running to the park

Why? Because there are alternatives which are much clearer:

1. He went running.
2. He went jogging, at the park.
3. He went to the park.
4. He went to the park, to have a run.
5. He went to the park, to go jogging.
6. He ran to the park.
7. He suddenly jumped up, and ran to the park.

It's difficult to imagine a case for "He went running to the park" but you are correct that it sounds ambiguous when considering those two meanings of "go". It could be either of the choices you mentioned, so it's better to avoid such a confusion.

P.S.

"go + Ving". Do you mean "go + ing"? That is, "going".