I was excited to give a shot to https://github.com/OpenNeuroDatasets/ds001544/ which has proper publicurl set etc... unfortunately there is something forbidding immediate parallel get (in below example -J8 but it is the same with -J2). It works on the 2nd try, or if not using -J. Error message states "unknown export location":

$> git annex version | head -n 1; git clone https://github.com/OpenNeuroDatasets/ds001544/ ; cd ds001544; git annex enableremote s3-PUBLIC; git annex get -J8 . 2>&1 | head -n 20; echo "2nd run"; git annex get -J8 .
git-annex version: 6.20181011+git7-g373c2abc2-1~ndall+1
Cloning into 'ds001544'...
remote: Enumerating objects: 866, done.
remote: Counting objects: 100% (866/866), done.
remote: Compressing objects: 100% (483/483), done.
remote: Total 866 (delta 144), reused 863 (delta 141), pack-reused 0
Receiving objects: 100% (866/866), 75.07 KiB | 4.69 MiB/s, done.
Resolving deltas: 100% (144/144), done.

(merging origin/git-annex into git-annex...)
(recording state in git...)

  Remote origin not usable by git-annex; setting annex-ignore
enableremote s3-PUBLIC ok
(recording state in git...)
get code/convert_sub01_ses01.R (from s3-PUBLIC...) 
  unknown export location

  Unable to access these remotes: s3-PUBLIC

  Try making some of these repositories available:
    1c66b8f9-34c7-42d1-8e9f-d7bc1982311a -- root@460f24a504cc:/datalad/ds001544
    837e28c7-9e4a-4792-b1b1-aa69d3430a42 -- [s3-PUBLIC]
    a7294efc-f620-445d-8e9d-803b3ec748ef -- s3-PRIVATE

  (Note that these git remotes have annex-ignore set: origin)
failed
get sub-01/ses-01/fmap/sub-01_ses-01_acq-cf0PA_run-03_epi.nii.gz (from s3-PUBLIC...) 
  unknown export location

  Unable to access these remotes: s3-PUBLIC

  Try making some of these repositories available:
    1c66b8f9-34c7-42d1-8e9f-d7bc1982311a -- root@460f24a504cc:/datalad/ds001544
    837e28c7-9e4a-4792-b1b1-aa69d3430a42 -- [s3-PUBLIC]
2nd run
get code/convert_sub01_ses02.R (from s3-PUBLIC...) (checksum...) ok
get code/convert_sub01_ses01.R (from s3-PUBLIC...) (checksum...) ok
get sub-01/ses-01/fmap/sub-01_ses-01_acq-cf1PA_run-02_epi.nii.gz (from s3-PUBLIC...) (checksum...) ok
get sub-01/ses-01/func/sub-01_ses-01_task-Stroop_acq-cf0AP_run-03_physio.tsv.gz (from s3-PUBLIC...) (checksum...) ok
...