If you are the petitioner and want the respondent to pay your costs then you will need to apply to the court for an order to make the respondent do so. It is important to let your solicitor know at the start of the proceedings if you wish to do this.
Going through a civil partnership dissolution is a very difficult time for both parties involved. It can be a time of high stress levels and can also be emotional. It may be worth balancing out how much upset asking the respondent to pay the legal costs would cause against how much the money would be worth to you.
It is sometimes the case that the court does not need to be involved in your financial arrangements with regards to your civil partnership dissolution if you are able to negotiate and agree to the terms between yourselves. If you were to claim the dissolution costs from the respondent through court then this could make them less willing to negotiate with you.
As always, it is important to get independent legal advice before making your decision.