I don't fully agree with stormy (second time this month).
Persimmon trees (kaki trees) are similar to peach trees, but not so extreme. They have strong apical growth, and quick death of non apical buds, so pruning is very important, or the tree will go out-of-control, without recovery options. (This is unlike apples).
What does this mean? The persimmon tree tends to make such "suckers" which grow tall, and to lose lower branches. You should prune many of these suckers, or shorten them, when they still have buds on lower part. Always look and make sure that there are enough buds and branches on the lower part, before you prune.
But contrary to peach trees, such suckers could still be useful. The fruits are heavy, and in general the tree tends to take a form similar to crying willows. These suckers will tend to bend down, except maybe the most vertical one, which will also have the stronger apical dominance, so prune them.
In this specific case, you have a lot of branches. I would prune many of the top ones, (keep the ones with better buds on the lower part of branch), and maybe also the lowest one if you have wild animals around you. This lowest branch will also bend down in autumn, and touch the ground.
In my experience, the branches are not very strong. They tend to break with snow, with heavy wind and with many fruits, so try to have fewer but stronger branches, and prune after the last chance of heavy snow.