Software as a Service (SaaS) applications fully exploit the potential of elastic Cloud computing Infrastructure as a Service (IaaS) platforms by enabling new highly dynamic Cloud provisioning scenarios where application providers could decide to change the placement of IT service components at runtime, such as moving computational resources close to storage so to improve SaaS responsiveness. Moreover, emergent Internet of Things (IoT) scenarios enable novel computing applications involving several heterogeneous smart objects interacting with each other. These highly dynamic scenarios call for novel Cloud support infrastructures able to automate the whole SaaS provisioning cycle spanning from resource management to dynamic IT service components placement, including software deployment, components re-activation, and rebinding operations. However, notwithstanding the core importance of these functions to truly enable the deployment of complex SaaS over IaaS environments, at the current stage only partial and ad-hoc solutions are available. This paper presents a support infrastructure aimed to facilitate the composition of heterogeneous resources, such as single Virtual Machines (VMs), DB services and storage, and stand-alone services, by automating the provisioning of complex SaaS applications over the widely diffused real-world open-source OpenStack IaaS. Collected experimental results show the effectiveness of parallel execution of deployment steps introduced by our solution and demonstrate its applicability and advantages in a real SaaS production testbed.