The DUP is a hard Brexit party.
No they aren't for the obvious reason that hard brexit would be likely to result in a hard border between NI and the republic being needed.
If you read their manifesto their key priorities on brexit are linked to trade and the economy and maintaining freedom of movement - that sounds pretty soft brexit to me. Specifically, their top priorities are:
DURING THE NEGOTIATIONS THE DUP WANTS TO SEE
A FOCUS ON THE FOLLOWING PRIORITIES AND OBJECTIVES:
1. Successful outward-looking knowledge-based economy in Northern Ireland
2. Ease of trade with the Irish Republic and throughout the European Union
3. Maintenance of the Common Travel Area
4. Strengthened relationships across the four components parts of the United Kingdom with no internal borders
5. Northern Ireland-specific solutions achieved through active Executive engagement
6. Particular circumstances of Northern Ireland with a land border with the EU fully reflected
7. Frictionless border with Irish Republic assisting those working or travelling in the other jurisdiction
8. Progress on new free trade deals with the rest of the world
9. Comprehensive free trade and customs agreement with the European Union
10. Northern Ireland established as a hub for trade from Irish Republic into the broader UK market
11. Customs arrangements which facilitate trade with new and existing markets
12. Confidence and capacity within local businesses to maximize opportunities
13. Business to retain competiveness and not face additional costs
14. Arrangements to facilitate ease of movement of people, goods and services
They don't want to restrict migration - quite the reverse (see 14), they are completely silent on single market, so no indication that they want to leave, and they want to retain opt-in access to various EU funding schemes.
Sounds all very soft-brexity to me.