This will depend a great deal on what your current qualifications are, where they are physically located and what jobs are available.
If you are located in the same geographic area, you probaly have about as good a chance as anyone else in the company for the types of jobs you are qualified for except you probably don't know the hiring managers yet which puts you at a disadvantage over people they know but not over people from outside the organization.
If you live outside the geographic area, you might be able to get a job if you are willing to move or they are willing to let you telecommute. However, they likely will not pay your moving expenses unless they are closing your office and you have to move to their HQ.
If you like the new parent and want to move to their other business, then start making personal contacts there. That will help more than anything to get you noticed by their hiring managers. If there are projects to integrate the two businesses, getting on the team to this will help you make those contacts. So volunteer.
One thing to be aware of is that business are often bought to get the customers not the employees of the company they bought. So start looking outside the company now even if they told you they had no plans to change anything (this is virtually 100% of the time a lie). If they don't have the same skill set, you may be safe, but.... let's just say I've seen this happen - a lot. Usually in about a year. Or they may outsource the jobs they don't do to another country and get rid of the expensive people who currently do them. The exception to this rule tends to be in the government contracting world where the contract usually included a set of skills required to have and may not allow outsourcing.