Re-land "Clean up the use of deprecated API in the analyzer_plugin package".

int.tryParse is now available to internal Google users, so it is safe
to land this change now.

This reverts commit 115850ca1d.
Original commit 86ba29265a.

Change-Id: I92c3d19533d3108c06910f21070266953c3a065b
Reviewed-on: https://dart-review.googlesource.com/53244
Reviewed-by: Konstantin Shcheglov <scheglov@google.com>
Commit-Queue: Paul Berry <paulberry@google.com>

pkg/analyzer_plugin/lib/src/protocol/protocol_internal.dart

@@ -277,9 +277,11 @@ abstract class JsonDecoder {
     if (json is int) {
       return json;
     } else if (json is String) {
-      return int.parse(json, onError: (String value) {
+      int value = int.tryParse(json);
+      if (value == null) {
         throw mismatch(jsonPath, 'int', json);
-      });
+      }
+      return value;
     }
     throw mismatch(jsonPath, 'int', json);
   }