Just like you, your parakeet doesn't like being cooped up in one place for too long. Unlike you, though, he spends all day locked in a cage. By letting him out, you help him relax, exercise and feel more comfortable around you, no matter the time of day.
The best out-of-cage time for a parakeet isn't a particular time of day, so long as it occurs every single day. Your parakeet doesn't necessarily care about what time it is; so as long as he's getting out at least once a day, he'll be happy. If you can, avoid going for particularly long stretches -- for example, if he gets out in the morning one day, don't wait until evening the next day to let him out again.
More Than Once
Ideally, your parakeet is getting out of his cage more than once a day. This is true especially whenever you're first training him, as he copes best with the stress of being handled in short bursts rather than extended training sessions. Think of it like walking the dog -- you should do it a few times a day for 15 minutes or so, and he'll get the hang of it. After he's fully used to being handled on a regular basis, you should let him out of the cage at least once a day, but you can make your sessions longer.
A Parakeet's Schedule
Parakeets enjoy getting plenty of sleep -- usually about 10-12 hours per day, not including the occasional midday nap. A typical parakeet gets the bulk of his sleep at night, just like you, so you don't have to worry about getting up at night to take him out of his cage. Some of the best times to take him out may be after he wakes up in the morning and whenever you get home in the evening. Don't wait until it's too late at night, though, or you'll be keeping him up past his bedtime.
Since there is no perfect time to take out your parakeet, all you can do is actually open the door and help him spread his wings. Hand-training your parakeet takes time, because face it -- you're a lot bigger than your bird, and in theory, you pose a pretty big threat. Don't take it personally if he's cautious about being handled, just keep it up a few times a day with consistency, and eventually he'll become more comfortable with leaving the cage and spending time with you.
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