It can be challenging, at least for me at first, to engage eclectus parrots in play. They seem to be a much more studious, watch every movement, hang on/near you, type creature, and have less of the "goof off" gene of a lot of other parrots have.
I felt like Matisse was missing out, so I started doing more physical things with him, that he hated in the beginning until he started to trust me more, and now has fun with the little "rough play" from time to time.
Once I got him comfortable with laying on his back, or hanging upside down from my fingers, we went from there. Now when I lay him on his back on the palm of my hand, he immediately releases my finger (before it was a death grip). I "throw him in the air like that, up and down, but not letting him go. I can tell he likes it (for a few times anyway), because he whistles and claps his wings afterwards and his pupils look like this (.)
He is clipped, but even when he wasn't he never flew unless he got scared. So I will hold him on my finger and say "wanna go flying?", he starts flapping his wings and I run back-and-forth in the house like a crazy person all while he screams and flaps his little heart out.
We go for short car rides through the neighboring neighborhoods, where he likes to wolf whistle at people walking their dogs, and say hello to people in the drive-through at the nearby McDonald's. I work from home, so we go usually during the weekdays or early in the morning on weekends where there's very little traffic, and I never go on any major roads or more than 2 miles from the house. He loves it so much, that the rewards outweigh the risk for me here. We sing to the radio with the windows cracked. Good times.
And of course there is shower fun. Let's just say sometimes my bathroom sounds like a rain forest.
Oh! I almost forgot our "risky business" game. I lightly hold his ankles/feet and slide him across the kitchen counter, letting go and he glides like the graceful butterfly he is. He'll run back one or two more times for another "ride", before he's over it.
Don't let anyone say that Ekkies are boring and just like to sit there. Sometimes you just have to break them out of their shell. The more you do with them, the more you can do with them.
And there are always training sessions. I use a target perch with him and do some training with treats and a clicker. He's learned to shake foot, turn around, puts his wings in the air when you point a finger and yell "freeze", give kisses, and plays dead, drops coins into a cup.
We also play tug of war with hair bands. I try to keep him active, because larger parrots can get cardiovascular issues from lack of exercise.