You could try a locking mechanism on the database instance to track parallel calls, something like the following:

private depth = 0;
private activeTransactions = 0; // tracks parallel transactions

async transaction<V>(done: (conn: this) => V): Promise<V> {
  if (this.activeTransactions > 0 && this.depth === 0) {
    //what to do if parallel transaction detected
  }

  this.activeTransactions += 1;
  this.depth += 1;

 await this.execute(`SAVEPOINT tt_${this.depth}`);
  try {
    return await done(this);
  } catch (err) {
    await this.execute(`ROLLBACK TO tt_${this.depth}`);
    throw err;
  } finally {
    await this.execute(`RELEASE tt_${this.depth}`);
    this.depth -= 1;
    this.activeTransactions -= 1;
  }
}

Alternatively, you could try using a WeakMap to track concurrent instances or you could use the async-mutex library to do the same thing.