I don't remember how Japan ordered the Parade of Nations when it hosted the Olympics in 1964 (when I was in high school there), but this year the nations were ordered according to how their Japanese names sounded in katakana, the Japanese syllabary used to render foreign names. A full list of the nations in Japanese order can be found in the NPR report about the parade.
Katakana order was used even when names contained kanji (Chinese characters). So Equatorial Guinea (赤道ギニア Sekidou Ginia, lit. 'Redroad [=equator] Guinea') appeared between Seychelles (セーシェル) and Senegal (セネガル) because they all start with the sound SE, written セ in katakana.
Similarly, Great Britain (英国 Eikoku, lit. 'brave-country') and the British Virgin Islands (英国ヴァージン諸島) appeared after Uruguay (ウルグァイ) and before Ecuador (エクアドル) because the katakana syllabary starts with the five vowels in the order A I U E O (アイウエオ), then proceeds to KA KI KU KE KO (カキクケコ). So the E+I of Eikoku precedes the E+KU of Ekuadoru. (In Chinese, where the name 英国 originated, the character 英 sounds much more like the first syllable of England.)
The last of the vowel-initial names are those that start with the sound O: Australia (オーストラリア Oosutoraria), Austria (オーストリア Oosutoria), Oman (オマーン Omaan), and the Netherlands (オランダ Oranda < Holland). I've transcribed the long vowels here as double vowels.
The order of the consonant-initial syllables is KA (カ), SA (サ), TA (タ), NA (ナ), HA (ハ), MA (マ), YA (ヤ), RA (ラ), WA (ワ), N (ン). Most, but not all, of these consonants occur with each vowel. The YA series has YA (ヤ), YU (ユ), and YO (ヨ), but YI and YE have been replaced by the vowels I and E. As a consequence, Yemen is written イェメン Iemen, and its team preceded Israel, Italy, Iraq, and Iran in the parade, while Jordan was relegated to near the end of the parade as the only name starting with Y, written ヨルダン Yorudan. The WA series only has WA (ワ) and WO (ヲ), with WI, WU, WE replaced by the vowels I, U, E. The final sound, N (ン) only occurs at the ends of syllables, as in Iemen and Yorudan.
In katakana, voiced consonants are distinguished from their voiced equivalents by a diacritic that looks a bit like a double quote mark: KA カ vs. GA ガ, TA タ vs. DA ダ, SA サ vs. ZA ザ. The consonants with and without diacritics are considered equivalent for ordering purposes. So Canada (Kanada), Gabon (Gabon), Cameroon (Kameruun), Gambia (Ganbia), Cambodia (Kanbojia) are in that order because of what follows their initial KA/GA syllables (-NA-, -BO-, -ME-, -NBI-, -NBO-, respectively). On the same principle, Zambia (Zanbia) precedes San Marino (Sanmarino) (-NBI- > -NMA-), while Singapore (Singaporu) precedes Zimbabwe (Zinbabue) (-NGA- > -NBA-) among the nations whose names start with S/Z.
The same principle applies to the three-way diacritical distinction between HA ハ, PA パ, and BA バ. So Bahrain (Baareen), Haiti (Haiti), and Pakistan (Pakisutan) begin the series of names beginning with HA ハ, which also include Vanuatu (Banuatu) because Japanese has no syllable VA. (However, the V can be represented by adding the voiced consonant diacritic " to the vowel ウ U, as in ヴァージン Vuaajin for the Virgin Islands.)
Nor does Japanese have a syllable FA, but the syllable HU (フ) sounds close enough to FU to substitute for F in foreign words. So names beginning with F sounds fall into the same group as those beginning with H, P, and B. Thus, the next countries to enter after Fiji (フィジー Fuijii), Philippines (フィリピン Fuiripin), and Finland (フィンァンド Fuinrando) were Bhutan (ブータン Buutan) and Puerto Rico (プエルトリコ Pueruto Riko).
The TA/DA (タ/ダ) series is at least as complicated. When pronounced, the syllables TA TI TU TE TO (タチツテト) actually sound like Ta Chi Tsu Te To and are usually romanized that way in English, while DA DI DU DE DO (ダヂヅデド) sound like Da Ji Zu De Do. So nations whose names start with Ch or Ts sounds are ordered among those whose names start with T/D. So the teams for Chile (Chiri), Tuvalu (Tsubaru), Denmark (Denmaaku), and Germany (Doitsu < Deutsch) entered in katakana order チツテト (TI TU TE TO, which sound like Chi, Tsu, Te, To), keeping in mind that TE=DE and TO=DO for ordering purposes.
Just as the normally syllabic フ FU can be combined with イ I (in フィ) to represent the foreign syllable FI, normally syllabic チ TI/CHI can be combined into チャ (TI+ya=) CHA, チュ (TI+yu=) CHU, チェ (TI+e=) CHE, and チョ (TI+yo =) CHO to represent foreign syllables starting with those sounds, as in チャイナ Chaina (China) or チェコ Cheko (Czech). Foreign words starting with J- can be represented using similar combinations starting with ZI/JI. So ZI+ya = JA in ジャマイカ Jamaica and ZI+yo = JO in ジョージア Georgia, which are sandwiched between ジブチ Djibouti and シリア Syria in katakana order. (Jordan is written ヨルダン Yorudan.)
It's interesting that the Republic of Korea, Chinese Taipei, and the People's Republic of China all appear among the nations whose names start with T/D, and the Democratic People's Republic of Korea would too, if it sent a team to this Olympics. The official name of South Korea in Chinese characters is 大韓民国 (Great Han Republic), which is pronounced in Japanese as Daikanminkoku. This name places South Korea immediately after Thailand (タイ Tai), which starts the T/D section of the parade of nations. Chinese Taipei (Chainiizu Taipei) and Tajikistan (Tajikisutan) immediately follow, so the former is ordered as if it were Taipei, not Chinese Taipei.
Tanzania, Czech (チェコ Cheko) Republic, Chad (チャド Chado), and the Central African Republic (中央アフリカ共和国 Chuuou Ahurika Kyouwakoku) precede China (中華人民共和国 Chuuka Jinmin Kyouwakoku 'Chinese ['Middle Splendor'] People's Republic') because the official names of both the CAR and PRC start with 中 'middle', which in katakana is written チュウ Chuu. The official name of North Korea in Chinese characters is 朝鮮民主主義人民共和国, pronounced in Japanese as Chousen Minshuushugi Jinmin Kyouwakoku ('Korean Democratic People's Republic'). It would immediately follow Tunisia (Chunijia) because チュ Chu precedes チョ Cho in katakana order.
Finally, because Japanese R renders both R and L in foreign names, and katakana RA RI RU RE RO come near the end of the syllabary, Laos, Latvia, Lithuania, Libya, Liechtenstein, Liberia, Romania (Ruumania), Luxembourg, Rwanda, Lesotho, and Lebanon come after Jordan (Yorudan) at the tail end of the parade, just before the current and future Olympic host nations.