Is the word parque the same in Brazilian and European Portuguese?

No.

While the core meaning of "parque" as a recreational area or an amusement park is identical in both dialects, there is a difference in typical usage regarding parking. In Brazil, the word for a parking lot is almost exclusively "estacionamento." In Portugal, it is standard to use "parque" as part of the compound term "parque de estacionamento." Additionally, the pronunciation differs: Brazilian Portuguese features more open vowels, whereas Continental Portuguese features more closed or reduced vowels (the final "e" is often nearly silent).

Brazilian Portuguese Examples:

1. Vamos ao parque amanhã? (Let's go to the park tomorrow?)
2. O parque de diversões é muito divertido. (The amusement park is very fun.)
3. O parque está muito cheio hoje. (The park is very crowded today.)
4. Vou levar meu cachorro para o parque. (I'm going to take my dog to the park.)
5. Preciso achar um estacionamento perto do parque. (I need to find a parking lot near the park.)

Portuguese (Portugal) Examples:

1. Vamos ao parque amanhã? (Let's go to the park tomorrow?)
2. O parque de diversões é muito giro. (The amusement park is very nice.)
3. O parque está muito cheio hoje. (The park is very crowded today.)
4. Vou levar o meu cão para o parque. (I'm going to take my dog to the park.)
5. Preciso encontrar um parque de estacionamento perto do parque. (I need to find a parking lot near the park.)