diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index c9a1dbd0ae90d..dfbbb456f50b6 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -348,8 +348,7 @@ Datetimelike
 Timedelta
 ^^^^^^^^^
 - Bug in :class:`TimedeltaIndex`, :class:`Series`, and :class:`DataFrame` floor-division with ``timedelta64`` dtypes and ``NaT`` in the denominator (:issue:`35529`)
--
--
+- Bug in parsing of ISO 8601 durations in :class:`Timedelta`, :meth:`pd.to_datetime` (:issue:`37159`, fixes :issue:`29773` and :issue:`36204`)
 
 Timezones
 ^^^^^^^^^
diff --git a/pandas/_libs/tslibs/timedeltas.pyx b/pandas/_libs/tslibs/timedeltas.pyx
index ee32ed53a908b..c6b47d09cf0bd 100644
--- a/pandas/_libs/tslibs/timedeltas.pyx
+++ b/pandas/_libs/tslibs/timedeltas.pyx
@@ -604,7 +604,7 @@ cdef inline int64_t parse_iso_format_string(str ts) except? -1:
 
     for c in ts:
         # number (ascii codes)
-        if ord(c) >= 48 and ord(c) <= 57:
+        if 48 <= ord(c) <= 57:
 
             have_value = 1
             if have_dot:
@@ -620,27 +620,28 @@ cdef inline int64_t parse_iso_format_string(str ts) except? -1:
             if not len(unit):
                 number.append(c)
             else:
-                # if in days, pop trailing T
-                if unit[-1] == 'T':
-                    unit.pop()
-                elif 'H' in unit or 'M' in unit:
-                    if len(number) > 2:
-                        raise ValueError(err_msg)
                 r = timedelta_from_spec(number, '0', unit)
                 result += timedelta_as_neg(r, neg)
 
                 neg = 0
                 unit, number = [], [c]
         else:
-            if c == 'P':
-                pass  # ignore leading character
+            if c == 'P' or c == 'T':
+                pass  # ignore marking characters P and T
             elif c == '-':
                 if neg or have_value:
                     raise ValueError(err_msg)
                 else:
                     neg = 1
-            elif c in ['D', 'T', 'H', 'M']:
+            elif c in ['W', 'D', 'H', 'M']:
                 unit.append(c)
+                if c in ['H', 'M'] and len(number) > 2:
+                    raise ValueError(err_msg)
+                r = timedelta_from_spec(number, '0', unit)
+                result += timedelta_as_neg(r, neg)
+
+                neg = 0
+                unit, number = [], []
             elif c == '.':
                 # append any seconds
                 if len(number):
@@ -661,11 +662,8 @@ cdef inline int64_t parse_iso_format_string(str ts) except? -1:
                     r = timedelta_from_spec(number, '0', dec_unit)
                     result += timedelta_as_neg(r, neg)
                 else:  # seconds
-                    if len(number) <= 2:
-                        r = timedelta_from_spec(number, '0', 'S')
-                        result += timedelta_as_neg(r, neg)
-                    else:
-                        raise ValueError(err_msg)
+                    r = timedelta_from_spec(number, '0', 'S')
+                    result += timedelta_as_neg(r, neg)
             else:
                 raise ValueError(err_msg)
 
diff --git a/pandas/tests/scalar/timedelta/test_constructors.py b/pandas/tests/scalar/timedelta/test_constructors.py
index 23fb25b838da6..06bdb8a6cf0a2 100644
--- a/pandas/tests/scalar/timedelta/test_constructors.py
+++ b/pandas/tests/scalar/timedelta/test_constructors.py
@@ -228,6 +228,14 @@ def test_overflow_on_construction():
         ("P0DT0H0M0.001S", Timedelta(milliseconds=1)),
         ("P0DT0H1M0S", Timedelta(minutes=1)),
         ("P1DT25H61M61S", Timedelta(days=1, hours=25, minutes=61, seconds=61)),
+        ("PT1S", Timedelta(seconds=1)),
+        ("PT0S", Timedelta(seconds=0)),
+        ("P1WT0S", Timedelta(days=7, seconds=0)),
+        ("P1D", Timedelta(days=1)),
+        ("P1DT1H", Timedelta(days=1, hours=1)),
+        ("P1W", Timedelta(days=7)),
+        ("PT300S", Timedelta(seconds=300)),
+        ("P1DT0H0M00000000000S", Timedelta(days=1)),
     ],
 )
 def test_iso_constructor(fmt, exp):
@@ -241,7 +249,6 @@ def test_iso_constructor(fmt, exp):
         "PDTHMS",
         "P0DT999H999M999S",
         "P1DT0H0M0.0000000000000S",
-        "P1DT0H0M00000000000S",
         "P1DT0H0M0.S",
     ],
 )