diff --git a/arrow/parser.py b/arrow/parser.py
index ee31a5bc..9ccf88df 100644
--- a/arrow/parser.py
+++ b/arrow/parser.py
@@ -107,7 +107,7 @@ class DateTimeParser:
     )
     _ESCAPE_RE: ClassVar[Pattern[str]] = re.compile(r"\[[^\[\]]*\]")
 
-    _ONE_OR_TWO_DIGIT_RE: ClassVar[Pattern[str]] = re.compile(r"\d{1,2}")
+    _ONE_OR_TWO_DIGIT_RE: ClassVar[Pattern[str]] = re.compile(r"[1-9]\d|\d")
     _ONE_OR_TWO_OR_THREE_DIGIT_RE: ClassVar[Pattern[str]] = re.compile(r"\d{1,3}")
     _ONE_OR_MORE_DIGIT_RE: ClassVar[Pattern[str]] = re.compile(r"\d+")
     _TWO_DIGIT_RE: ClassVar[Pattern[str]] = re.compile(r"\d{2}")
diff --git a/tests/test_parser.py b/tests/test_parser.py
index bdcc1026..ee121af3 100644
--- a/tests/test_parser.py
+++ b/tests/test_parser.py
@@ -147,7 +147,7 @@ def test_YY_and_YYYY_format_list(self):
 
         assert self.parser.parse(
             "15/01/2019T04:05:06.789120Z",
-            ["D/M/YYThh:mm:ss.SZ", "D/M/YYYYThh:mm:ss.SZ"],
+            ["D/MM/YYThh:mm:ss.SZ", "D/MM/YYYYThh:mm:ss.SZ"],
         ) == datetime(2019, 1, 15, 4, 5, 6, 789120, tzinfo=tz.tzutc())
 
     # regression test for issue #447
@@ -790,6 +790,23 @@ def test_parse_normalize_whitespace(self):
         with pytest.raises(ParserError):
             self.parser.parse("  \n Jun   1\t 2005\n ", "MMM D YYYY")
 
+    def test_parse_leading_zero(self):
+        # Regression tests for issue #1084
+        with pytest.raises(ParserError):
+            self.parser.parse("01", "M")
+
+        with pytest.raises(ParserError):
+            self.parser.parse("01", "D")
+
+        with pytest.raises(ParserError):
+            self.parser.parse("01", "H")
+
+        with pytest.raises(ParserError):
+            self.parser.parse("01", "h")
+
+        with pytest.raises(ParserError):
+            self.parser.parse("01", "s")
+
 
 @pytest.mark.usefixtures("dt_parser_regex")
 class TestDateTimeParserRegex: