diff --git a/src/basic/time-util.c b/src/basic/time-util.c
index 7d44358cd8..657e231b0d 100644
--- a/src/basic/time-util.c
+++ b/src/basic/time-util.c
@@ -662,6 +662,9 @@ static int parse_timestamp_impl(
          *   +5min
          *   -5days
          *   @2147483647          (seconds since epoch)
+         *
+         * Note, on DST change, 00:00:00 may not exist and in that case the time part may be shifted.
+         * E.g. "Sun 2023-03-13 America/Havana" is parsed as "Sun 2023-03-13 01:00:00 CDT".
          */
 
         assert(t);
diff --git a/src/basic/time-util.h b/src/basic/time-util.h
index 087f5324ef..79a6715656 100644
--- a/src/basic/time-util.h
+++ b/src/basic/time-util.h
@@ -205,13 +205,17 @@ static inline usec_t usec_sub_signed(usec_t timestamp, int64_t delta) {
         return usec_sub_unsigned(timestamp, (usec_t) delta);
 }
 
+/* The last second we can format is 31. Dec 9999, 1s before midnight, because otherwise we'd enter 5 digit
+ * year territory. However, since we want to stay away from this in all timezones we take one day off. */
+#define USEC_TIMESTAMP_FORMATTABLE_MAX_64BIT ((usec_t) 253402214399000000) /* Thu 9999-12-30 23:59:59 UTC */
+/* With a 32bit time_t we can't go beyond 2038...
+ * We parse timestamp with RFC-822/ISO 8601 (e.g. +06, or -03:00) as UTC, hence the upper bound must be off
+ * by USEC_PER_DAY. See parse_timestamp() for more details. */
+#define USEC_TIMESTAMP_FORMATTABLE_MAX_32BIT (((usec_t) INT32_MAX) * USEC_PER_SEC - USEC_PER_DAY)
 #if SIZEOF_TIME_T == 8
-  /* The last second we can format is 31. Dec 9999, 1s before midnight, because otherwise we'd enter 5 digit
-   * year territory. However, since we want to stay away from this in all timezones we take one day off. */
-#  define USEC_TIMESTAMP_FORMATTABLE_MAX ((usec_t) 253402214399000000)
+#  define USEC_TIMESTAMP_FORMATTABLE_MAX USEC_TIMESTAMP_FORMATTABLE_MAX_64BIT
 #elif SIZEOF_TIME_T == 4
-/* With a 32bit time_t we can't go beyond 2038... */
-#  define USEC_TIMESTAMP_FORMATTABLE_MAX ((usec_t) 2147483647000000)
+#  define USEC_TIMESTAMP_FORMATTABLE_MAX USEC_TIMESTAMP_FORMATTABLE_MAX_32BIT
 #else
 #  error "Yuck, time_t is neither 4 nor 8 bytes wide?"
 #endif
diff --git a/src/test/test-date.c b/src/test/test-date.c
index 097066b61a..16c6d4e5be 100644
--- a/src/test/test-date.c
+++ b/src/test/test-date.c
@@ -101,8 +101,8 @@ int main(int argc, char *argv[]) {
         test_should_fail("9999-12-31 00:00:00 UTC");
         test_should_fail("10000-01-01 00:00:00 UTC");
 #elif SIZEOF_TIME_T == 4
-        test_should_pass("2038-01-19 03:14:07 UTC");
-        test_should_fail("2038-01-19 03:14:08 UTC");
+        test_should_pass("2038-01-18 03:14:07 UTC");
+        test_should_fail("2038-01-18 03:14:08 UTC");
 #endif
 
         return 0;
diff --git a/src/test/test-time-util.c b/src/test/test-time-util.c
index e974520bbe..e0386d4ec5 100644
--- a/src/test/test-time-util.c
+++ b/src/test/test-time-util.c
@@ -407,6 +407,10 @@ static void test_format_timestamp_impl(usec_t x) {
 
 static void test_format_timestamp_loop(void) {
         test_format_timestamp_impl(USEC_PER_SEC);
+        test_format_timestamp_impl(USEC_TIMESTAMP_FORMATTABLE_MAX_32BIT-1);
+        test_format_timestamp_impl(USEC_TIMESTAMP_FORMATTABLE_MAX_32BIT);
+        test_format_timestamp_impl(USEC_TIMESTAMP_FORMATTABLE_MAX-1);
+        test_format_timestamp_impl(USEC_TIMESTAMP_FORMATTABLE_MAX);
 
         for (unsigned i = 0; i < TRIAL; i++) {
                 usec_t x;
@@ -617,8 +621,8 @@ TEST(format_timestamp_range) {
         test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX + 1, TIMESTAMP_US_UTC, "--- XXXX-XX-XX XX:XX:XX.XXXXXX UTC");
         test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX + 1, TIMESTAMP_DATE, "--- XXXX-XX-XX");
 #elif SIZEOF_TIME_T == 4
-        test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX, TIMESTAMP_UTC, "Tue 2038-01-19 03:14:07 UTC");
-        test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX, TIMESTAMP_DATE, "Tue 2038-01-19");
+        test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX, TIMESTAMP_UTC, "Mon 2038-01-18 03:14:07 UTC");
+        test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX, TIMESTAMP_DATE, "Mon 2038-01-18");
         test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX + 1, TIMESTAMP_UTC, "--- XXXX-XX-XX XX:XX:XX UTC");
         test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX + 1, TIMESTAMP_US_UTC, "--- XXXX-XX-XX XX:XX:XX.XXXXXX UTC");
         test_format_timestamp_one(USEC_TIMESTAMP_FORMATTABLE_MAX + 1, TIMESTAMP_DATE, "--- XXXX-XX-XX");
@@ -632,12 +636,30 @@ static void test_parse_timestamp_one(const char *str, usec_t max_diff, usec_t ex
         int r;
 
         r = parse_timestamp(str, &usec);
-        log_debug("/* %s(%s): max_diff="USEC_FMT", expected="USEC_FMT", result="USEC_FMT"*/", __func__, str, max_diff, expected, usec);
+        log_debug("/* %s(%s): max_diff="USEC_FMT", expected="USEC_FMT", result="USEC_FMT" */", __func__, str, max_diff, expected, usec);
         assert_se(r >= 0);
         assert_se(usec >= expected);
         assert_se(usec_sub_unsigned(usec, expected) <= max_diff);
 }
 
+static bool time_is_zero(usec_t usec) {
+        const char *s;
+
+        s = FORMAT_TIMESTAMP(usec);
+        return strstr(s, " 00:00:00 ");
+}
+
+static bool timezone_equal(usec_t today, usec_t target) {
+        const char *s, *t, *sz, *tz;
+
+        s = FORMAT_TIMESTAMP(today);
+        t = FORMAT_TIMESTAMP(target);
+        assert_se(sz = strrchr(s, ' '));
+        assert_se(tz = strrchr(t, ' '));
+        log_debug("%s("USEC_FMT", "USEC_FMT") -> %s, %s", __func__, today, target, s, t);
+        return streq(sz, tz);
+}
+
 static void test_parse_timestamp_impl(const char *tz) {
         usec_t today, now_usec;
 
@@ -819,12 +841,17 @@ static void test_parse_timestamp_impl(const char *tz) {
 
         /* without date */
         assert_se(parse_timestamp("today", &today) == 0);
-        test_parse_timestamp_one("00:01", 0, today + USEC_PER_MINUTE);
-        test_parse_timestamp_one("00:00:01", 0, today + USEC_PER_SEC);
-        test_parse_timestamp_one("00:00:01.001", 0, today + USEC_PER_SEC + 1000);
-        test_parse_timestamp_one("00:00:01.0010", 0, today + USEC_PER_SEC + 1000);
-        test_parse_timestamp_one("tomorrow", 0, today + USEC_PER_DAY);
-        test_parse_timestamp_one("yesterday", 0, today - USEC_PER_DAY);
+        if (time_is_zero(today)) {
+                test_parse_timestamp_one("00:01", 0, today + USEC_PER_MINUTE);
+                test_parse_timestamp_one("00:00:01", 0, today + USEC_PER_SEC);
+                test_parse_timestamp_one("00:00:01.001", 0, today + USEC_PER_SEC + 1000);
+                test_parse_timestamp_one("00:00:01.0010", 0, today + USEC_PER_SEC + 1000);
+
+                if (timezone_equal(today, today + USEC_PER_DAY) && time_is_zero(today + USEC_PER_DAY))
+                        test_parse_timestamp_one("tomorrow", 0, today + USEC_PER_DAY);
+                if (timezone_equal(today, today - USEC_PER_DAY) && time_is_zero(today - USEC_PER_DAY))
+                        test_parse_timestamp_one("yesterday", 0, today - USEC_PER_DAY);
+        }
 
         /* relative */
         assert_se(parse_timestamp("now", &now_usec) == 0);