Assumed role in AWS Lambda, access denied on SSM call

I'm getting an error in my Lambda function, which calls SSM:

AccessDeniedException: User: arn:aws:sts::redacted:assumed-role/LambdaBackend_master_lambda/SpikeLambda is not authorized to perform: ssm:GetParameter on resource: arn:aws:ssm:eu-west-1:redacted:parameter/default/key/api

However, I'm pretty sure I configured this correctly:

Role, with AssumeRole for Lambda (although we know that works from the error message).

λ aws iam get-role --role-name LambdaBackend_master_lambda
{
    "Role": {
        "AssumeRolePolicyDocument": {
            "Version": "2012-10-17",
            "Statement": [
                {
                    "Action": "sts:AssumeRole",
                    "Effect": "Allow",
                    "Principal": {
                        "Service": "lambda.amazonaws.com"
                    }
                }
            ]
        },
        "RoleId": "redacted",
        "CreateDate": "2017-06-23T20:49:37Z",
        "RoleName": "LambdaBackend_master_lambda",
        "Path": "/",
        "Arn": "arn:aws:iam::redacted:role/LambdaBackend_master_lambda"
    }
}

And my policy:

λ aws iam list-role-policies --role-name LambdaBackend_master_lambda
{
    "PolicyNames": [
        "ssm_read"
    ]
}
λ aws iam get-role-policy --role-name LambdaBackend_master_lambda --policy-name ssm_read
{
    "RoleName": "LambdaBackend_master_lambda",
    "PolicyDocument": {
        "Version": "2012-10-17",
        "Statement": [
            {
                "Action": [
                    "ssm:DescribeParameters"
                ],
                "Resource": "*",
                "Effect": "Allow"
            },
            {
                "Action": [
                    "ssm:GetParameters"
                ],
                "Resource": "arn:aws:ssm:eu-west-1:redacted:parameter/*",
                "Effect": "Allow"
            }
        ]
    },
    "PolicyName": "ssm_read"
}

I've run it through the policy simulator and it seems to be fine!

Played around with this today and got the following, dropping the s from ssm:GetParameters and using ssm:GetParameter seems to work when using the GetParameter action. ie AWS_PROFILE=pstore aws ssm get-parameter --name param_name. This weirded me out a bit because I cannot find this at all in the iam action docs here. However it does seem to work, and ssm is still a bit under documented. Amazon has updated and moved it's docs. The new docs incude both ssm:GetParameters and ssm:GetParameter.

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Action": [
                "ssm:DescribeParameters"
            ],
            "Resource": "*",
            "Effect": "Allow"
        },
        {
            "Action": [
                "ssm:GetParameter"
            ],
            "Resource": "arn:aws:ssm:eu-west-1:redacted:parameter/*",
            "Effect": "Allow"
        }
    ]
}