diff --git a/pandas/io/html.py b/pandas/io/html.py
index 490c574463b9b..715f0c08546a3 100644
--- a/pandas/io/html.py
+++ b/pandas/io/html.py
@@ -897,7 +897,7 @@ def _parse(flavor, io, match, attrs, encoding, displayed_only, **kwargs):
 
         try:
             tables = p.parse_tables()
-        except Exception as caught:
+        except ValueError as caught:
             # if `io` is an io-like object, check if it's seekable
             # and try to rewind it before trying the next parser
             if hasattr(io, "seekable") and io.seekable():
diff --git a/pandas/io/parsers.py b/pandas/io/parsers.py
index 3678e32943b2e..2d8303632e723 100755
--- a/pandas/io/parsers.py
+++ b/pandas/io/parsers.py
@@ -3271,24 +3271,29 @@ def converter(*date_cols):
                 )
         else:
             try:
-                result = tools.to_datetime(
-                    date_parser(*date_cols), errors="ignore", cache=cache_dates
-                )
-                if isinstance(result, datetime.datetime):
-                    raise Exception("scalar parser")
-                return result
+                parsed_cols = date_parser(*date_cols)
+                if isinstance(parsed_cols, datetime.datetime):
+                    raise TypeError("scalar parser")
             except Exception:
+                # Since `date_parser` is user-provided, we can't guess
+                #  what it might raise.
+                dcs = parsing._concat_date_cols(date_cols)
                 try:
-                    return tools.to_datetime(
-                        parsing.try_parse_dates(
-                            parsing._concat_date_cols(date_cols),
-                            parser=date_parser,
-                            dayfirst=dayfirst,
-                        ),
-                        errors="ignore",
+                    parsed = parsing.try_parse_dates(
+                        dcs, parser=date_parser, dayfirst=dayfirst
                     )
                 except Exception:
+                    # Since `date_parser` is user-provided, we can't guess
+                    #  what it might raise.
                     return generic_parser(date_parser, *date_cols)
+                else:
+                    return tools.to_datetime(parsed, errors="ignore")
+
+            else:
+                result = tools.to_datetime(
+                    parsed_cols, errors="ignore", cache=cache_dates
+                )
+                return result
 
     return converter