Rename parser dir to parse (#3178)

Continuing with #3070. Just a dir and file rename (mostly removing
prefixes, although for parse_tree_fuzzer and parse_tree_file_test I'm
dropping "tree" instead of "parse"). Everything in the parse dir should
be marked as a move.

toolchain/parse/parse_file_test.cpp

@@ -0,0 +1,26 @@
+// Part of the Carbon Language project, under the Apache License v2.0 with LLVM
+// Exceptions. See /LICENSE for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+
+#include <string>
+
+#include "llvm/ADT/SmallVector.h"
+#include "toolchain/driver/driver_file_test_base.h"
+
+namespace Carbon::Testing {
+namespace {
+
+class ParseFileTest : public DriverFileTestBase {
+ public:
+  using DriverFileTestBase::DriverFileTestBase;
+
+  auto GetDefaultArgs() -> llvm::SmallVector<std::string> override {
+    return {"compile", "--phase=parse", "--dump-parse-tree", "%s"};
+  }
+};
+
+}  // namespace
+
+CARBON_FILE_TEST_FACTORY(ParseFileTest);
+
+}  // namespace Carbon::Testing