In the superproject, right after a clone, run these commands:

#!/bin/sh
#define the actions to take when running submodule update for each submodule
repo_root=$(git rev-parse --show-toplevel)
git config submodule.sub1.update '!'"${repo_root}/sparse1.sh"
git config submodule.sub2.update '!'"${repo_root}/sparse2.sh"

In this example, we need two scripts because we want different dirs checked out in sub1 and sub2.
See "dir/to/checkout" in script example below.
These scripts will be run in lieu of the normal "git checkout <SHA1>" command that is run as part of the submodule update command.

Then, you can run the update with init command as usual:

git submodule update --init


Contents of the the sparse checkout scripts (sparse1.sh / sparse2.sh):

#!/bin/sh
#script is called with a single argument, the hash of the submodule commit recorded in the superproject,
#and it runs in the submodule directory
git config core.sparseCheckout true
sub_git_dir=$(git rev-parse --git-dir)
echo dir/to/checkout > ${sub_git_dir}/info/sparse-checkout
git checkout --quiet $1