Add this extension method
public static class ForEachAsyncExtension
{
    public static Task ForEachAsync<T>(this IEnumerable<T> source, int dop, Func<T, Task> body)
    {
        return Task.WhenAll(from partition in Partitioner.Create(source).GetPartitions(dop) 
            select Task.Run(async delegate
            {
                using (partition)
                    while (partition.MoveNext())
                        await body(partition.Current).ConfigureAwait(false);
            }));
    }
}
And then use like so:
Task.Run(async () =>
{
    var s3 = new AmazonS3Client(Config.Instance.Aws.Credentials, Config.Instance.Aws.RegionEndpoint);
    var buckets = await s3.ListBucketsAsync();
    foreach (var s3Bucket in buckets.Buckets)
    {
        if (s3Bucket.BucketName.StartsWith("mybucket-"))
        {
            log.Information("Bucket => {BucketName}", s3Bucket.BucketName);
            ListObjectsResponse objects;
            try
            {
                objects = await s3.ListObjectsAsync(s3Bucket.BucketName);
            }
            catch
            {
                log.Error("Error getting objects. Bucket => {BucketName}", s3Bucket.BucketName);
                continue;
            }
            // ForEachAsync (4 is how many tasks you want to run in parallel)
            await objects.S3Objects.ForEachAsync(4, async s3Object =>
            {
                try
                {
                    log.Information("Bucket => {BucketName} => {Key}", s3Bucket.BucketName, s3Object.Key);
                    await s3.DeleteObjectAsync(s3Bucket.BucketName, s3Object.Key);
                }
                catch
                {
                    log.Error("Error deleting bucket {BucketName} object {Key}", s3Bucket.BucketName, s3Object.Key);
                }
            });
            try
            {
                await s3.DeleteBucketAsync(s3Bucket.BucketName);
            }
            catch
            {
                log.Error("Error deleting bucket {BucketName}", s3Bucket.BucketName);
            }
        }
    }
}).Wait();