Answer

Here is one commonly printed order of the 108 names of Shri Parshvanath from the Parshvanath Ashtottarashatanamavali. I am keeping the names in sequence as they appear in the stotra. ( sanskritdocuments.org)

Explanation

1–18

1. Jinaya
2. Paramashankaraya
3. Nathaya
4. Paramashaktaye
5. Sharanyaya
6. Sarva Kamadaya
7. Sarva Vighnaharaya
8. Swamine
9. Siddhipadapradayakaya
10. Sarva Sattvahitaya
11. Yogine
12. Shri Karaya
13. Paramarthadaya
14. Devadevaya
15. Swayamsiddhaya
16. Chidanandamayaya
17. Shivaya
18. Paramatmane. ( sanskritdocuments.org)

19–36

1. Parabrahmane
2. Paramaya
3. Parameshvaraya
4. Jagannathaya
5. Surajyeshthaya
6. Bhuteshaya
7. Purushottamaya
8. Surendraya
9. Nityadharmaya
10. Shrinivasaya
11. Sudarnavaya
12. Sarvajnaya
13. Sarvadeveshaya
14. Sarvagaya
15. Sarvatomukhaya
16. Sarvatmane
17. Sarvadarshine
18. Sarvavyapine. ( sanskritdocuments.org)

37–54

1. Jagadgurave
2. Tattvamurtaye
3. Paradityaya
4. Parabrahmaprakashaya
5. Paramendave
6. Parapranaya
7. Paramamrita Siddhidaya
8. Ajaya
9. Sanatanaya
10. Shambhave
11. Ishvaraya
12. Sadashivaya
13. Vishveshvaraya
14. Pramodatmane
15. Kshetradhishaya
16. Shubhapradaya
17. Sakaraya
18. Nirakaraya. ( sanskritdocuments.org)

55–72

1. Sakalaya
2. Nishkalaya
3. Avyayaya
4. Nirmamaya
5. Nirvikaraya
6. Nirvikalpaya
7. Niramayaya
8. Amaraya
9. Arujaya
10. Anantaya
11. Ekaya
12. Anekaya
13. Shivatmakaya
14. Alakshyaya
15. Aprameyaya
16. Dhyanalakshyaya
17. Niranjanaya
18. Karakritaye. ( sanskritdocuments.org)

73–90

1. Avyaktaya
2. Vyaktarupaya
3. Trayimayaya
4. Brahmadwayaya
5. Prakashatmane
6. Nirbhayaya
7. Paramaksharaya
8. Divya Tejomayaya
9. Shantaya
10. Paramamritamayaya
11. Achyutaya
12. Aghaya
13. Anadyaya
14. Pareshanaya
15. Parameshthine
16. Parah Pumanse
17. Shuddha Sphatika Sankashaya
18. Swayambhuve. ( sanskritdocuments.org)

91–108

1. Paramachyutaya
2. Vyomakarasvarupaya
3. Lokayalokavabhasakaya
4. Jnanatmane
5. Пaramanandaya
6. Pranarudhaya
7. Manahsthitaye
8. Manah Sadhyaya
9. Mano Dhyeyaya
10. Manodrishyaya
11. Paraparaya
12. Sarvatirthamayaya
13. Nityaya
14. Sarvadevamayaya
15. Prabhave
16. Bhagavate
17. Sarvatatveshaya
18. Shivashrisaukhyadayakaya. ( sanskritdocuments.org)

Takeaway

This is the traditional Ashtottarashatanamavali sequence for Shri Parshvanath. If you want, I can also format it as a clean chanting list or give it in Devanagari script.