For example, I read that brush park was an historic neighborhood close to downtown, but then I see that much of it has been torn down or is in ruins like Athens or Rome. I also saw "a tour of Detroit's ghetto" on YouTube and could not believe it. Is most of the city like this? How is this possible in the united states?
Most of the city is not like the YouTube "tours of the ghetto." The point of those is to find the most torn up neighborhoods. The majority of the city is not vacant.

The United States has a long history of economic ghost towns. Detroit has economic ghost neighborhoods. Much of the housing could be considered company housing that sprouted up around factories during the postwar boom. That industry has since globalized and Detroit hasn't been able to completely fill in the vacuum. Half the population has left since the peak in the 50's. More than 1.8 million to more than 700K. The metro area's population has been stagnant since the 70s [[and actually it recently declined slightly). But there are a lot of engineering jobs here.

Am I being naive or are things really as bad as some make them out to be?
I mean you were wise enough to ask so I wouldn't call you naive. The situation is bad from a governmental standpoint as it looks as if the city government is about to run out of money. The situation is alright from a people standpoint as there have been more people living in the downtown areas where development strategies have been focused.

Brush Park is part of a strange band between Midtown & Downtown that we're still trying to figure out what to do with. The freeways cut it up and there are some social services that aren't being managed properly to ensure the safety of the surrounding neighborhood. But we might just put an ice rink there and we just built some low income housing.

The situation in the city neighborhoods can go either way. There will be blocks with beautiful mansions or stable older stock homes and then burnt out blocks in the process of being demolished or whole blocks that are just grass that get cut by the city or volunteers/neighborhood groups.

So the positive spin I can put on all that is Detroit is well on its way to having even more abundant parks.