Across:
1 Bridge (CROSS) notices (SECTION-) rewritten (= anagram indicator) for school (-S) (CROSS SECTIONS) with numerous representatives (CROSS SECTIONS),
10 An option to update (REFRESH) silly (= anagram indicator) Fresher (fresher = REFRESH),
11 Oriental (east = E-) reader (-LECTOR) (ELECTOR) gets to decide who goes to the park? (ELECTOR gets to decide who goes to The Park/presidency),
12 Ocean-liner (SAND) going around Sao Tiago Island (going around “Sao Tiago Island” = SAND),
13 Strike out (CROSS out) for all sides today (CROSS on all sides of today’s grid),
15 Previous night’s (EVES) sample of herringbone vests (sample of “herringbone vests” = EVES),
17 Parody (APE) of those in Budapest (those in “Budapest” = APE),
19 Listen secretly (EARWIG), going on ahead (on a head = -WIG) after (= position indicator) a sound type (EAR-) (EARWIG),
21 Discovers (DIGS UP) jobs for musicians (gigs) beginning to go (g) (“gigs” without “g” = -IGS) in party up north (Democratic Unionist Party = DUP = D- UP) (DIGS UP),
22 27 across (“toe” = TOE-) is hammered (-NAIL) (TOENAIL) and gets clipped by one of those associated with The Polish (TOENAIL gets clipped + is associated with nail polish),
23 Turns up (= reversal indicator), graduate (BA =A-B) hides revenue (ROS = -SOR-) from director (-D-) (ADSORB) – on the surface, could hold water? (ADSORB),
25 Harry (harry/hound = PESTER) Street (-ST-) fenced in by foreign aristocrat (peer = PE-ER) (PESTER),
27 Characters in Dostoevsky’s (characters in “Dostoevsky’s” = TOE) Tests the Water (TOE),
29 Locked up in Guggenheim – a museum (locked up in “Guggenheim – a museum” = IMAM) preaching to the faithful (IMAM),
30 Taste (SENSE) sample of delicatessen seasonings (sample of “delicatessen seasonings” = SENSE),
31 Cover by unpatriotic (cover of “unpatriotic” = TRIO) group of musicians (TRIO),
34 Away (= anagram indicator) goalie (goalie = E-LOGIA) catches unbelievable header (“unbelievable” header/first letter = -U-) (EULOGIA) getting the accolades (EULOGIA),
35 A (A-) conservative (-C-) made (made = -ADEM-) out (= anagram indicator) Spain (-E) (ACADEME) appeals to those in study (ACADEME),
36 Switch sides – politically speaking (CROSS THE FLOOR) with one of those in The Park (CROSS) and most of (m) them (“them” without “m” = THE) on the level (FLOOR) (CROSS THE FLOOR).
Down:
2 Authoritative type from Ring (referee in boxing ring = REF-) in (-IN-) hospital ward (-ER) (REFINER) removing foreign bodies perhaps (removing foreign bodies/impurities = REFINER),
3 Catch sight of (SEE-) first drummer (first “drummer” = -D) (SEED) with Genesis (genesis/origin = SEED),
4 Has (has = SAH-) come up (= reversal indicator) with a (-A-) Trim (y) chipper regular (“ray” without “y” = -RA) (SAHARA) for a hot, dry African spread (SAHARA),
5 Encourages a grin (say CHEESE) and that’s coming from the posers (CHEESE),
6 Thought (IDEA) that came from some outside agency (that came from some “outside agency” = IDEA),
7 Sent via (sent via = NATIVES) train (= anagram indicator) by locals (NATIVES),
8 Does a job at bar (CROSS-EXAMINES) offering snappy (CROSS-) grills (-EXAMINES) (CROSS-EXAMINES),
9 Initial cap (initial “cap” = C-) on revenue (Revenue Online Service = -ROS-) from endless (t) shoot (“spurt” without “t” = -S-PUR) with models (-POSES) (CROSS-PURPOSES) leads to mix-ups (CROSS-PURPOSES),
14 In my view (OPINION), there’ll be tears over this (onion = O-NION) involving detective (-PI-) (OPINION),
16 Man taken out of terminal (“man” taken out of “terminal” = teril = LITRE) has capacity to hold less than two pints (LITRE = 1.75 pints/less than two pints),
18 Serves up (= reversal indicator) restaurant (deli = -ILED) order (O-) (OILED) that’s covered in grease (OILED),
20 Local mouthy type (GOB) found in San Domingo bar (found in “San Domingo bar” = GOB),
21 Bit of Bacardi punch (bit of “Bacardi punch” = DIP) for those who like their sauce (DIP),
24 Gone off (STA-LE-) to cover initial posting (initial “posting” = -P-) with resistance (-R) (STAPLER) and grips the papers in one go (STAPLER),
26 In retrospect (= reversal indicator), old (-O-) railway heading (“railway” heading =-R-) out of train station (depot = TO-PED) (TORPEDO) is explosive for those in the shipping game (TORPEDO),
27 Fragment of overwritten anthology (fragment of “overwritten anthology” = TENANT) on the landlord’s books (TENANT),
28 Questionable (= anagram indicator) space (space = ESCAP-) for sweetheart (“sweet” heart = -E) (ESCAPE) to get free (ESCAPE),
32 Tried to abandon tragedies (“tried” to abandon “tragedies” = ages = AGES) for a long time (AGES),
33 War is hell for the banks (“war is hell” for the banks = WALL) in Berlin (Berlin WALL).