diff --git a/sqlglot/dialects/hive.py b/sqlglot/dialects/hive.py
index 089e6aa75..cc7debb66 100644
--- a/sqlglot/dialects/hive.py
+++ b/sqlglot/dialects/hive.py
@@ -319,7 +319,9 @@ class Parser(parser.Parser):
             "TO_DATE": build_formatted_time(exp.TsOrDsToDate, "hive"),
             "TO_JSON": exp.JSONFormat.from_arg_list,
             "UNBASE64": exp.FromBase64.from_arg_list,
-            "UNIX_TIMESTAMP": build_formatted_time(exp.StrToUnix, "hive", True),
+            "UNIX_TIMESTAMP": lambda args: build_formatted_time(exp.StrToUnix, "hive", True)(
+                args or [exp.CurrentTimestamp()]
+            ),
             "YEAR": lambda args: exp.Year(this=exp.TsOrDsToDate.from_arg_list(args)),
         }
 
diff --git a/sqlglot/dialects/presto.py b/sqlglot/dialects/presto.py
index 7b3103466..c1580e346 100644
--- a/sqlglot/dialects/presto.py
+++ b/sqlglot/dialects/presto.py
@@ -26,6 +26,7 @@
     timestrtotime_sql,
     ts_or_ds_add_cast,
 )
+from sqlglot.dialects.hive import Hive
 from sqlglot.dialects.mysql import MySQL
 from sqlglot.helper import apply_index_offset, seq_get
 from sqlglot.tokens import TokenType
@@ -404,9 +405,6 @@ class Generator(generator.Generator):
             exp.StrToDate: lambda self, e: f"CAST({_str_to_time_sql(self, e)} AS DATE)",
             exp.StrToMap: rename_func("SPLIT_TO_MAP"),
             exp.StrToTime: _str_to_time_sql,
-            exp.StrToUnix: lambda self, e: self.func(
-                "TO_UNIXTIME", self.func("DATE_PARSE", e.this, self.format_time(e))
-            ),
             exp.StructExtract: struct_extract_sql,
             exp.Table: transforms.preprocess([_unnest_sequence]),
             exp.Timestamp: no_timestamp_sql,
@@ -439,6 +437,22 @@ class Generator(generator.Generator):
             exp.Xor: bool_xor_sql,
         }
 
+        def strtounix_sql(self, expression: exp.StrToUnix) -> str:
+            # Since `TO_UNIXTIME` requires a `TIMESTAMP`, we need to parse the argument into one.
+            # To do this, we first try to `DATE_PARSE` it, but since this can fail when there's a
+            # timezone involved, we wrap it in a `TRY` call and use `PARSE_DATETIME` as a fallback,
+            # which seems to be using the same time mapping as Hive, as per:
+            # https://joda-time.sourceforge.net/apidocs/org/joda/time/format/DateTimeFormat.html
+            value_as_text = exp.cast(expression.this, "text")
+            parse_without_tz = self.func("DATE_PARSE", value_as_text, self.format_time(expression))
+            parse_with_tz = self.func(
+                "PARSE_DATETIME",
+                value_as_text,
+                self.format_time(expression, Hive.INVERSE_TIME_MAPPING, Hive.INVERSE_TIME_TRIE),
+            )
+            coalesced = self.func("COALESCE", self.func("TRY", parse_without_tz), parse_with_tz)
+            return self.func("TO_UNIXTIME", coalesced)
+
         def bracket_sql(self, expression: exp.Bracket) -> str:
             if expression.args.get("safe"):
                 return self.func(
diff --git a/sqlglot/generator.py b/sqlglot/generator.py
index b3a56ae8d..5aee11834 100644
--- a/sqlglot/generator.py
+++ b/sqlglot/generator.py
@@ -3192,11 +3192,16 @@ def format_args(self, *args: t.Optional[str | exp.Expression]) -> str:
     def text_width(self, args: t.Iterable) -> int:
         return sum(len(arg) for arg in args)
 
-    def format_time(self, expression: exp.Expression) -> t.Optional[str]:
+    def format_time(
+        self,
+        expression: exp.Expression,
+        inverse_time_mapping: t.Optional[t.Dict[str, str]] = None,
+        inverse_time_trie: t.Optional[t.Dict] = None,
+    ) -> t.Optional[str]:
         return format_time(
             self.sql(expression, "format"),
-            self.dialect.INVERSE_TIME_MAPPING,
-            self.dialect.INVERSE_TIME_TRIE,
+            inverse_time_mapping or self.dialect.INVERSE_TIME_MAPPING,
+            inverse_time_trie or self.dialect.INVERSE_TIME_TRIE,
         )
 
     def expressions(
diff --git a/tests/dialects/test_dialect.py b/tests/dialects/test_dialect.py
index 4c3cf5d2c..3c3a1c62e 100644
--- a/tests/dialects/test_dialect.py
+++ b/tests/dialects/test_dialect.py
@@ -611,7 +611,7 @@ def test_time(self):
             write={
                 "duckdb": "EPOCH(STRPTIME('2020-01-01', '%Y-%m-%d'))",
                 "hive": "UNIX_TIMESTAMP('2020-01-01', 'yyyy-MM-dd')",
-                "presto": "TO_UNIXTIME(DATE_PARSE('2020-01-01', '%Y-%m-%d'))",
+                "presto": "TO_UNIXTIME(COALESCE(TRY(DATE_PARSE(CAST('2020-01-01' AS VARCHAR), '%Y-%m-%d')), PARSE_DATETIME(CAST('2020-01-01' AS VARCHAR), 'yyyy-MM-dd')))",
                 "starrocks": "UNIX_TIMESTAMP('2020-01-01', '%Y-%m-%d')",
                 "doris": "UNIX_TIMESTAMP('2020-01-01', '%Y-%m-%d')",
             },
diff --git a/tests/dialects/test_hive.py b/tests/dialects/test_hive.py
index 3acf2e918..04326d300 100644
--- a/tests/dialects/test_hive.py
+++ b/tests/dialects/test_hive.py
@@ -369,7 +369,7 @@ def test_time(self):
             "UNIX_TIMESTAMP(x)",
             write={
                 "duckdb": "EPOCH(STRPTIME(x, '%Y-%m-%d %H:%M:%S'))",
-                "presto": "TO_UNIXTIME(DATE_PARSE(x, '%Y-%m-%d %T'))",
+                "presto": "TO_UNIXTIME(COALESCE(TRY(DATE_PARSE(CAST(x AS VARCHAR), '%Y-%m-%d %T')), PARSE_DATETIME(CAST(x AS VARCHAR), 'yyyy-MM-dd HH:mm:ss')))",
                 "hive": "UNIX_TIMESTAMP(x)",
                 "spark": "UNIX_TIMESTAMP(x)",
                 "": "STR_TO_UNIX(x, '%Y-%m-%d %H:%M:%S')",
diff --git a/tests/dialects/test_spark.py b/tests/dialects/test_spark.py
index 1ac13fbac..88c9ab4b4 100644
--- a/tests/dialects/test_spark.py
+++ b/tests/dialects/test_spark.py
@@ -246,12 +246,15 @@ def test_spark(self):
         self.validate_identity("SELECT TRANSFORM(ARRAY(1, 2, 3), (x, i) -> x + i)")
         self.validate_identity("REFRESH TABLE a.b.c")
         self.validate_identity("INTERVAL -86 DAYS")
-        self.validate_identity("SELECT UNIX_TIMESTAMP()")
         self.validate_identity("TRIM('    SparkSQL   ')")
         self.validate_identity("TRIM(BOTH 'SL' FROM 'SSparkSQLS')")
         self.validate_identity("TRIM(LEADING 'SL' FROM 'SSparkSQLS')")
         self.validate_identity("TRIM(TRAILING 'SL' FROM 'SSparkSQLS')")
         self.validate_identity("SPLIT(str, pattern, lim)")
+        self.validate_identity(
+            "SELECT UNIX_TIMESTAMP()",
+            "SELECT UNIX_TIMESTAMP(CURRENT_TIMESTAMP())",
+        )
         self.validate_identity(
             "SELECT CAST('2023-01-01' AS TIMESTAMP) + INTERVAL 23 HOUR + 59 MINUTE + 59 SECONDS",
             "SELECT CAST('2023-01-01' AS TIMESTAMP) + INTERVAL '23' HOUR + INTERVAL '59' MINUTE + INTERVAL '59' SECONDS",
diff --git a/tests/test_transpile.py b/tests/test_transpile.py
index f6fd2f9a7..0349b9d74 100644
--- a/tests/test_transpile.py
+++ b/tests/test_transpile.py
@@ -702,7 +702,11 @@ def test_time(self):
         )
 
         self.validate("STR_TO_TIME('x', 'y')", "DATE_PARSE('x', 'y')", write="presto")
-        self.validate("STR_TO_UNIX('x', 'y')", "TO_UNIXTIME(DATE_PARSE('x', 'y'))", write="presto")
+        self.validate(
+            "STR_TO_UNIX('x', 'y')",
+            "TO_UNIXTIME(COALESCE(TRY(DATE_PARSE(CAST('x' AS VARCHAR), 'y')), PARSE_DATETIME(CAST('x' AS VARCHAR), 'y')))",
+            write="presto",
+        )
         self.validate("TIME_TO_STR(x, 'y')", "DATE_FORMAT(x, 'y')", write="presto")
         self.validate("TIME_TO_UNIX(x)", "TO_UNIXTIME(x)", write="presto")
         self.validate(