diff --git a/pandas/_libs/tslibs/parsing.pyi b/pandas/_libs/tslibs/parsing.pyi
index db1388672b37c..c7244447edaf7 100644
--- a/pandas/_libs/tslibs/parsing.pyi
+++ b/pandas/_libs/tslibs/parsing.pyi
@@ -23,7 +23,7 @@ def _does_string_look_like_datetime(py_string: str) -> bool: ...
 def quarter_to_myear(year: int, quarter: int, freq: str) -> tuple[int, int]: ...
 def try_parse_dates(
     values: npt.NDArray[np.object_],  # object[:]
-    parser=...,
+    parser,
     dayfirst: bool = ...,
     default: datetime | None = ...,
 ) -> npt.NDArray[np.object_]: ...
diff --git a/pandas/_libs/tslibs/parsing.pyx b/pandas/_libs/tslibs/parsing.pyx
index 992e1d90f4f3b..6d834c5494a83 100644
--- a/pandas/_libs/tslibs/parsing.pyx
+++ b/pandas/_libs/tslibs/parsing.pyx
@@ -661,7 +661,7 @@ cdef dateutil_parse(
 
 
 def try_parse_dates(
-    object[:] values, parser=None, bint dayfirst=False, default=None,
+    object[:] values, parser, bint dayfirst=False, default=None,
 ) -> np.ndarray:
     cdef:
         Py_ssize_t i, n
@@ -670,32 +670,11 @@ def try_parse_dates(
     n = len(values)
     result = np.empty(n, dtype="O")
 
-    if parser is None:
-        if default is None:  # GH2618
-            date = datetime.now()
-            default = datetime(date.year, date.month, 1)
-
-        def parse_date(x):
-            return du_parse(x, dayfirst=dayfirst, default=default)
-
-        # EAFP here
-        try:
-            for i in range(n):
-                if values[i] == "":
-                    result[i] = np.nan
-                else:
-                    result[i] = parse_date(values[i])
-        except Exception:
-            # Since parser is user-defined, we can't guess what it might raise
-            return values
-    else:
-        parse_date = parser
-
-        for i in range(n):
-            if values[i] == "":
-                result[i] = np.nan
-            else:
-                result[i] = parse_date(values[i])
+    for i in range(n):
+        if values[i] == "":
+            result[i] = np.nan
+        else:
+            result[i] = parser(values[i])
 
     return result.base  # .base to access underlying ndarray
 
diff --git a/pandas/io/parsers/base_parser.py b/pandas/io/parsers/base_parser.py
index 9fdb1380e14eb..f38666ac52529 100644
--- a/pandas/io/parsers/base_parser.py
+++ b/pandas/io/parsers/base_parser.py
@@ -1121,19 +1121,13 @@ def converter(*date_cols):
         if date_parser is None:
             strs = parsing.concat_date_cols(date_cols)
 
-            try:
-                return tools.to_datetime(
-                    ensure_object(strs),
-                    utc=False,
-                    dayfirst=dayfirst,
-                    errors="ignore",
-                    cache=cache_dates,
-                ).to_numpy()
-
-            except ValueError:
-                return tools.to_datetime(
-                    parsing.try_parse_dates(strs, dayfirst=dayfirst), cache=cache_dates
-                )
+            return tools.to_datetime(
+                ensure_object(strs),
+                utc=False,
+                dayfirst=dayfirst,
+                errors="ignore",
+                cache=cache_dates,
+            ).to_numpy()
         else:
             try:
                 result = tools.to_datetime(
diff --git a/pandas/tests/io/parser/test_parse_dates.py b/pandas/tests/io/parser/test_parse_dates.py
index bb1619c3d7bde..080410b0c913a 100644
--- a/pandas/tests/io/parser/test_parse_dates.py
+++ b/pandas/tests/io/parser/test_parse_dates.py
@@ -165,7 +165,9 @@ def date_parser(*date_cols):
         -------
         parsed : Series
         """
-        return parsing.try_parse_dates(parsing.concat_date_cols(date_cols))
+        return parsing.try_parse_dates(
+            parsing.concat_date_cols(date_cols), parser=du_parse
+        )
 
     kwds = {
         "header": None,
@@ -2026,3 +2028,31 @@ def test_parse_dates_and_string_dtype(all_parsers):
     expected = DataFrame({"a": ["1"], "b": [Timestamp("2019-12-31")]})
     expected["a"] = expected["a"].astype("string")
     tm.assert_frame_equal(result, expected)
+
+
+def test_parse_dot_separated_dates(all_parsers):
+    # https://github.com/pandas-dev/pandas/issues/2586
+    parser = all_parsers
+    data = """a,b
+27.03.2003 14:55:00.000,1
+03.08.2003 15:20:00.000,2"""
+    if parser.engine == "pyarrow":
+        expected_index = Index(
+            ["27.03.2003 14:55:00.000", "03.08.2003 15:20:00.000"],
+            dtype="object",
+            name="a",
+        )
+        warn = None
+    else:
+        expected_index = DatetimeIndex(
+            ["2003-03-27 14:55:00", "2003-08-03 15:20:00"],
+            dtype="datetime64[ns]",
+            name="a",
+        )
+        warn = UserWarning
+    msg = "when dayfirst=False was specified"
+    result = parser.read_csv_check_warnings(
+        warn, msg, StringIO(data), parse_dates=True, index_col=0
+    )
+    expected = DataFrame({"b": [1, 2]}, index=expected_index)
+    tm.assert_frame_equal(result, expected)
diff --git a/pandas/tests/tslibs/test_parsing.py b/pandas/tests/tslibs/test_parsing.py
index 2cfb7393ed96c..7675305e27d22 100644
--- a/pandas/tests/tslibs/test_parsing.py
+++ b/pandas/tests/tslibs/test_parsing.py
@@ -4,7 +4,7 @@
 from datetime import datetime
 import re
 
-from dateutil.parser import parse
+from dateutil.parser import parse as du_parse
 import numpy as np
 import pytest
 
@@ -271,9 +271,11 @@ def test_guess_datetime_format_no_padding(string, fmt, dayfirst, warning):
 
 def test_try_parse_dates():
     arr = np.array(["5/1/2000", "6/1/2000", "7/1/2000"], dtype=object)
-    result = parsing.try_parse_dates(arr, dayfirst=True)
+    result = parsing.try_parse_dates(
+        arr, dayfirst=True, parser=lambda x: du_parse(x, dayfirst=True)
+    )
 
-    expected = np.array([parse(d, dayfirst=True) for d in arr])
+    expected = np.array([du_parse(d, dayfirst=True) for d in arr])
     tm.assert_numpy_array_equal(result, expected)