diff --git a/compactor/src/parquet_file_combining.rs b/compactor/src/parquet_file_combining.rs
index fcc05e353b..157bca1d61 100644
--- a/compactor/src/parquet_file_combining.rs
+++ b/compactor/src/parquet_file_combining.rs
@@ -3,6 +3,7 @@ use data_types::{
     CompactionLevel, ParquetFile, ParquetFileId, ParquetFileParams, PartitionId, TableSchema,
 };
 use datafusion::error::DataFusionError;
+use futures::{stream::FuturesOrdered, StreamExt, TryStreamExt};
 use iox_catalog::interface::Catalog;
 use iox_query::{
     exec::{Executor, ExecutorType},
@@ -44,6 +45,9 @@ pub(crate) enum Error {
     #[snafu(display("Error executing compact plan  {}", source))]
     ExecuteCompactPlan { source: DataFusionError },
 
+    #[snafu(display("Error executing parquet write task  {}", source))]
+    ExecuteParquetTask { source: tokio::task::JoinError },
+
     #[snafu(display("Could not serialize and persist record batches {}", source))]
     Persist {
         source: parquet_file::storage::UploadError,
@@ -215,60 +219,89 @@ pub(crate) async fn compact_parquet_files(
         .await
         .context(CompactPhysicalPlanSnafu)?;
 
+    let partition = Arc::new(partition);
+
     // Run to collect each stream of the plan
     let stream_count = physical_plan.output_partitioning().partition_count();
-    let mut compacted_parquet_files = Vec::with_capacity(stream_count);
 
     debug!("running plan with {} streams", stream_count);
-    for i in 0..stream_count {
-        trace!(partition = i, "executing datafusion partition");
-        let data = ctx
-            .execute_stream_partitioned(Arc::clone(&physical_plan), i)
-            .await
-            .context(ExecuteCompactPlanSnafu)?;
-        trace!(partition = i, "built result stream for partition");
 
-        let meta = IoxMetadata {
-            object_store_id: Uuid::new_v4(),
-            creation_timestamp: time_provider.now(),
-            sequencer_id: partition.sequencer_id(),
-            namespace_id: partition.namespace_id(),
-            namespace_name: partition.namespace.name.clone().into(),
-            table_id: partition.table.id,
-            table_name: partition.table.name.clone().into(),
-            partition_id,
-            partition_key: partition.partition_key.clone(),
-            max_sequence_number,
-            compaction_level: CompactionLevel::FileNonOverlapped,
-            sort_key: Some(sort_key.clone()),
-        };
+    // These streams *must* to run in parallel otherwise a deadlock
+    // can occur. Since there is a merge in the plan, in order to make
+    // progress on one stream there must be (potential space) on the
+    // other streams.
+    //
+    // https://github.com/influxdata/influxdb_iox/issues/4306
+    // https://github.com/influxdata/influxdb_iox/issues/4324
+    let compacted_parquet_files = (0..stream_count)
+        .map(|i| {
+            // Prepare variables to pass to the closure
+            let ctx = exec.new_context(ExecutorType::Reorg);
+            let physical_plan = Arc::clone(&physical_plan);
+            let store = store.clone();
+            let time_provider = Arc::clone(&time_provider);
+            let sort_key = sort_key.clone();
+            let partition = Arc::clone(&partition);
+            // run as a separate tokio task so files can be written
+            // concurrently.
+            tokio::task::spawn(async move {
+                trace!(partition = i, "executing datafusion partition");
+                let data = ctx
+                    .execute_stream_partitioned(physical_plan, i)
+                    .await
+                    .context(ExecuteCompactPlanSnafu)?;
+                trace!(partition = i, "built result stream for partition");
 
-        debug!(
-            ?partition_id,
-            "executing and uploading compaction StreamSplitExec"
-        );
+                let meta = IoxMetadata {
+                    object_store_id: Uuid::new_v4(),
+                    creation_timestamp: time_provider.now(),
+                    sequencer_id: partition.sequencer_id(),
+                    namespace_id: partition.namespace_id(),
+                    namespace_name: partition.namespace.name.clone().into(),
+                    table_id: partition.table.id,
+                    table_name: partition.table.name.clone().into(),
+                    partition_id,
+                    partition_key: partition.partition_key.clone(),
+                    max_sequence_number,
+                    compaction_level: CompactionLevel::FileNonOverlapped,
+                    sort_key: Some(sort_key.clone()),
+                };
 
-        let object_store_id = meta.object_store_id;
-        info!(?partition_id, %object_store_id, "streaming exec to object store");
+                debug!(
+                    ?partition_id,
+                    "executing and uploading compaction StreamSplitExec"
+                );
 
-        // Stream the record batches from the compaction exec, serialize
-        // them, and directly upload the resulting Parquet files to
-        // object storage.
-        let (parquet_meta, file_size) = store.upload(data, &meta).await.context(PersistSnafu)?;
+                let object_store_id = meta.object_store_id;
+                info!(?partition_id, %object_store_id, "streaming exec to object store");
 
-        debug!(?partition_id, %object_store_id, "file uploaded to object store");
+                // Stream the record batches from the compaction exec, serialize
+                // them, and directly upload the resulting Parquet files to
+                // object storage.
+                let (parquet_meta, file_size) =
+                    store.upload(data, &meta).await.context(PersistSnafu)?;
 
-        let parquet_file = meta.to_parquet_file(partition_id, file_size, &parquet_meta, |name| {
-            partition
-                .table_schema
-                .columns
-                .get(name)
-                .expect("unknown column")
-                .id
-        });
+                debug!(?partition_id, %object_store_id, "file uploaded to object store");
 
-        compacted_parquet_files.push(parquet_file);
-    }
+                let parquet_file =
+                    meta.to_parquet_file(partition_id, file_size, &parquet_meta, |name| {
+                        partition
+                            .table_schema
+                            .columns
+                            .get(name)
+                            .expect("unknown column")
+                            .id
+                    });
+
+                Ok(parquet_file)
+            })
+        })
+        // NB: FuturesOrdered allows the futures to run in parallel
+        .collect::<FuturesOrdered<_>>()
+        // Check for errors in the task
+        .map(|t| t.context(ExecuteParquetTaskSnafu)?)
+        .try_collect::<Vec<_>>()
+        .await?;
 
     update_catalog(
         catalog,